Walks and Lines
Chris Godsil
June 22, 2013
Preface
We introduce two topics where combinatorics and quantum physics interact.
iii
Contents
Preface
iii
Contents
v
1 Continuous Quantum Walks
1.1 Basics . . . . . . . . . . . . . . . . . . . . . . .
1.2 Some Physics . . . . . . . . . . . . . . . . . . .
1.3 Products . . . . . . . . . . . . . . . . . . . . .
1.4 State Transfer and Mixing . . . . . . . . . . .
1.5 Spectral Decomposition for Adjacency Matrices
1.6 Using Spectral Decomposition . . . . . . . . .
1.7 Bounding Transfer . . . . . . . . . . . . . . . .
1.8 Bipartite Graphs . . . . . . . . . . . . . . . . .
1.9 Vertex-Transitive Graphs . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
2
3
4
5
6
7
9
10
12
2 State Transfer
2.1 Three Inequalities . . . . . . .
2.2 Symmetry and Periodicity . . .
2.3 Periods . . . . . . . . . . . . .
2.4 Bounding the Minimum Period
2.5 Transition Operators and Walk
2.6 Integrality . . . . . . . . . . .
2.7 No Control . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
15
15
17
18
20
21
22
26
3 Real
3.1
3.2
3.3
. . . . .
. . . . .
. . . . .
. . . . .
Modules
. . . . .
. . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Lines
29
Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Equiangular Lines . . . . . . . . . . . . . . . . . . . . . . . 30
The Relative Bound . . . . . . . . . . . . . . . . . . . . . . 32
v
Contents
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
Gram Matrices . . . . . . . . .
Number Theory . . . . . . . .
Switching . . . . . . . . . . . .
Paley Graphs . . . . . . . . . .
A Spherical 2-Design . . . . .
An Example . . . . . . . . . .
Graphs from Equiangular Lines
Distance-Regular Graphs . . .
Strongly Regular Graphs . . .
SRG’s: Examples . . . . . . .
Equitable Partitions . . . . . .
Some Algebra . . . . . . . . .
Inner Products . . . . . . . . .
Positive Semidefinite Matrices
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
34
36
37
38
39
41
41
43
45
46
47
49
50
51
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
53
53
54
55
56
58
60
5 Mutually Unbiased Bases
5.1 Basics . . . . . . . . . . . . . . . . . . .
5.2 Bounds . . . . . . . . . . . . . . . . . .
5.3 MUB’s . . . . . . . . . . . . . . . . . .
5.4 Real MUB’s . . . . . . . . . . . . . . .
5.5 Cayley Graphs and Incidence Structures
5.6 Difference Sets . . . . . . . . . . . . . .
5.7 Difference Sets and Equiangular Lines .
5.8 Relative Difference Sets and MUB’s . .
5.9 Type-II Matrices over Abelian Groups .
5.10 Difference Sets and Equiangular Lines .
5.11 Affine Planes . . . . . . . . . . . . . . .
5.12 Products . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
63
63
64
64
66
68
69
70
71
72
73
74
75
4 Complex Lines
4.1 The Absolute Bound . .
4.2 The Relative Bound . .
4.3 Gram Matrices . . . . .
4.4 Type-II Matrices . . . .
4.5 The Unitary Group . .
4.6 An Extraspecial Group
Bibliography
vi
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
77
Contents
Index
79
vii
Chapter 1
Continuous Quantum Walks
A continuous random walk on a graph X can be specified by a family of
matrices M (t), where M (t)a,b is the probability that at time t the “walker”
is on vertex b, given that it started at vertex a. If A is the adjacency matrix
of X and ∆ is the diagonal matrix such that ∆a,a is the valency of a, then
we can assume that
M (t) = exp(t(A − ∆)).
The underlying model is that in a short time interval of length δt, the walker
moves to an adjacent vertex with probability proportional to δt, and it is
equally likely to move to any neighbor of the vertex it is on. We note that
each column of M (t) is a probability density—its entries are nonnegative
and sum to 1.
In their work on quantum computing, physicists have introduced continuous quantum walks, which we can define as follows. Assume A is a real
symmetric matrix and define the transition operator U (t) by
U (t) := exp(itA).
Generally we want A to be sparse and the cases of most interest to us
arise when A is the adjacency matrix or the Laplacian of a graph X. The
matrix A is the Hamiltonian of the walk. Physicists most frequently use the
adjacency matrix and so this will be our default assumption. We observe
that U (t) = U (−t) and (since A is symmetric) that U (t)T = U (t); hence
U (t)∗ U (t) = U (−t)U (t) = I,
equivalently U (t) is a unitary matrix for all t. (For a complex matrix U , we
T
are using U ∗ to denote the conjugate-transpose U of U .)
1
1. Continuous Quantum Walks
Since U (t) is unitary, the norm of each row and column is 1. Let M (t)
denote the matrix we get by replacing each entry of U (t) by the square of its
absolute value, we will call it the mixing matrix of the walk. Each column
of M (t) is a probability density, and so it is not unreasonable to view U (t)
as giving rise to a continuous quantum walk.
1.1
Basics
By way of example, take X = K2 and A = A(K2 ). Then
0 1
A=
1 0
!
and so
A2k = I,
A2k+1 = A.
Therefore
1
1
1
exp(itA) = I + itA − t2 I − it3 A + t4 A + · · ·
2
6
24
= cos(t)I + i sin(t)A
and
!
cos(t) i sin(t)
U (t) =
.
i sin(t) cos(t)
At particular times t, this matrix takes a special form. Thus
U (π) = −I,
while
0 1
U (π/2) = i
1 0
and
!
!
1 1 i
U (π/4) = √
.
2 i 1
Thus U (t) is respectively:
(a) A scalar matrix.
(b) A scalar times a permutation matrix.
2
1.2. Some Physics
(c) A unitary matrix with all entries having the same absolute value.
For any graph X, since A is symmetric, the operator U (t) is symmetric.
Since iA is Hermitian,
U (t) = exp(−itA) = U (−t),
and hence U (t) is unitary for all t.
1.2
Some Physics
The operator U (t) determines the evolution of a quantum system. For us
a quantum system will be a finite-dimensional vector space V over C, with
the inner product
hx, yi := x∗ y.
A state of a quantum system is a 1-dimensional subspace, thus it is a point
in the projective space associated with V. The usual practice in physics is
to denote a line by a unit vector that spans it. If x and y are unit vectors
that span the same line, there is a complex scalar γ of norm 1 such that
y = γx. Physicists refer to γ as a phase factor.
We set up (or prepare) a quantum system by applying a unitary operator
to it. To perform a measurement on a quantum system, we must first
choose an orthonormal basis. If dim(V) = d, then it is customary to use
the standard basis e1 , . . . , ed . The result of a measurement is an element of
{1, . . . , d}; if our system is in a state represented by a unit vector z when
we measure it, the outcome is r with probability
|her , zi|2 .
For our purposes it is useful to note that this probability is the r-th entry
of the vector z ◦ z. It is crucial to note that this Schur product contains
all the information that can be obtained by measurement—in general no
measurement will provide the value of the r-th entry zr of z. Equivalently
zr is determined only up to a phase factor.
If the initial state of the system is the standard basis vector ea and we apply our quantum walk operator U (t) to it, then the result of a measurement
relative to the standard basis is determined by the vector
U (t)ea ◦ U (t)ea = U (t)ea ◦ U (−t)ea
3
1. Continuous Quantum Walks
As mere mathematicians, we can avoid thinking about the underlying physics
by simply making sure that we restrict ourselves to questions about the vectors U (t)ea ◦ U (−t)ea . (This will not guarantee that the question will be
physically interesting, but at least it will have a physical meaning.)
The mixing matrix M (t) is given by
M (t) = U (t) ◦ U (−t),
For a, b ∈ V (X), the entry M (t)a,b is the probability that at time t the
system is in the state eb , given that its initial state was ea . For the path on
two vertices we have
!
cos2 (t) sin2 (t)
M (t) =
.
sin2 (t) cos2 (t)
Clearly the mixing matrix is doubly stochastic, and each column is a probability density. We will be most interested in determining if there is a time
t such that a column of M (t) takes a specified form: for example, whether
some entry is equal to 1 (in which case all other entries are 0), or whether
all entries are equal.
1.3
Products
The Kronecker product A ⊗ B of matrix A and B is the matrix we get if,
for each i and j we replace the ij-entry of A by Ai,j B. So if A is k × ` and
B is m × n, the product A ⊗ B is a matrix of order km × `n. Note that if
` = n = 1 then we can view A ⊗ B is an element of Ckm . Hence we have
vector spaces V1 and V2 we can define their tensor product to be the span of
the vectors v1 ⊗ v2 , for all v1 in V1 and v2 in V2 . It is an axiom of quantum
physics that if V1 and V2 are the state space of quantum systems, the state
space of the composite system is V1 ⊗ V2 .
The crucial property of the Kronecker product is that, if the products
AC and BD are defined, then
(A ⊗ B)(C ⊗ D) = AC ⊗ BD.
(For example, A ⊗ B = (A ⊗ I)(I ⊗ B).) The Kronecker product provides a
convenient way to define products of graphs, in particular if X and Y are
4
1.4. State Transfer and Mixing
graphs with respective adjacency matrices A and B, the adjacency matrix
of their Cartesian product X Y is
A ⊗ I + I ⊗ B.
1.3.1 Lemma. If X and Y are graphs, then
UXY = UX ⊗ UY .
Proof. Assume the adjacency matrices of X and Y are A and B respectively.
The matrices A ⊗ I and I ⊗ B commute, whence
exp(it(A ⊗ I + I ⊗ B)) = exp(it(A ⊗ I)) exp(it(I ⊗ B))
= (exp(itA) ⊗ I)(I ⊗ exp(itB))
= exp(itA) ⊗ exp(itB).
It follows, for example, that the transition matrix for the d-cube is the
d-th tensor power of the transition matrix for P2 .
UP2 (t)⊗d .
If L(X) denotes the Laplacian of X then you are invited to verify that
L(X Y ) = (L(X) ⊗ I) + (I ⊗ L(Y ))
Hence whether we use the adjacency matrix or the Laplacian, the above
lemma holds.
The Kronecker product interacts nicely with Schur product, thus if A
and C have the same order and B and D have the same order,
(A ⊗ B) ◦ (C ⊗ D) = (A ◦ C) ⊗ (B ◦ D).
From this it follows that
MXY (t) = MX (t) ⊗ MY (t).
1.4
State Transfer and Mixing
Suppose A is a symmetric matrix, which we view as a weighted adjacency
matrix of a graph X. (Hence it could be an adjacency matrix, a signed
5
1. Continuous Quantum Walks
adjacency matrix, or a Laplacian.) We say that we have perfect state
transfer from vertex u in X to vertex v at time t if there is a complex
scalar γ such that
U (t)eu = γev .
Since U (t) is unitary, the vectors eu and γev have the same length and
consequently |γ| = 1—it is a phase factor, as discussed in Section 1.2. From
Section 1.1 we see that if A = A(K2 ), we have perfect state transfer from
vertex 1 to vertex 2 at time π/2. From Section 1.3 it follows that if u is a
vertex in the Cartesian power K2d , then at time π/2 we have perfect state
transfer from u to the unique vertex at distance d from u. (Note that K2d
is better known as the d-cube.)
We say that we have instantaneous uniform mixing, or just uniform
mixing at time t if all entries of U (t) have the same absolute value. (We
say that matrix with this property is flat.) For K2 we saw that U (π/4) is
flat. Hence we have uniform mixing at time π/4 on K2 and, more generally,
on the d-cube Qd .
Our two basic problems are to determine the cases where perfect state
transfer occurs, and to determine when uniform mixing occurs.
1.5
Spectral Decomposition for Adjacency
Matrices
Suppose A is a symmetric matrix with distinct eigenvalues θ1 , . . . , θm and
let Ei denote orthogonal projection onto the eigenspace belonging to θi .
Then Ei is a polynomial in A,
X
Ei = I
i
and
Ei = Ei = EiT
and
Ei Ej = δi,j Ei .
Further
A=
X
i
6
θi Ei ;
1.6. Using Spectral Decomposition
this identity is known as the spectral decomposition of A. If f is a function
defined on the spectrum of A, then
f (A) =
X
f (θi )Ei .
i
All this is standard, see [4, Section 3.5] for example.
One case of particular interest to us is the following identity:
(tI − A)−1 =
X
i
1
Ei .
t − θi
This allows us to express walk generating functions on a graph X in terms
of its eigenvalues and the entries of the idempotents Ei . Thus
φ(X \ u, t) X (Ei )u,u
=
.
φ(X, t)
t − θi
i
We will take this up at length in Chapter ??.
By way of example, we compute the pieces in the spectral decomposition
of P3 . The characteristic polynomial of P3 is t3 − 2t, whence its eigenvalues
are
√
√
2, 0, − 2.
The idempotents Ei (in the same
√



1
2 √1
1
√
1
1

2 ,
 2
 0
√2
4
2
−1
1
2 1
order) are

0 −1
0 0
,
0 1
√

1
− 2
1
√
√
1

2
− 2 .
− 2
√
4
1
− 2
1

(In computing these it helps to note that if an eigenvalue is simple and z
is an associated eigenvector with norm 1, then the projection E is equal to
zz T .)
1.6
Using Spectral Decomposition
So far the only graphs we have considered are K2 and its Cartesian powers.
To increase our range, we make extensive use of the spectral decomposition
of symmetric matrices. If A is symmetric and has spectral decomposition
A=
X
θr Er
r
7
1. Continuous Quantum Walks
then
U (t) = exp(itA) =
X
eiθr t Er .
r
Thus U (t) is a polynomial in A for any t, and its eigenvalues are the numbers
eiθr t .
By way of example, we apply this to the path P3 . The eigenvalues of P3
are
√
√
2, 0, − 2,
we denote the corresponding idempotents by E1 , E2 and E3 . Since the
eigenvalues are simple, each idempotent is of the form zz T , where z is an
eigenvector of norm 1. The normalized eigenvectors are


1
1 √ 
 2 ,
2
1


1
1  
√  0 ,
2 −1
and consequently our idempotents are
√




1
2
1
1
0
−1
√ 
1
1 √

2
2
 2
,
 0 0 0 ,
√
4
2
−1 0 1
1
2 1
We have
U (t) = ei
and therefore
√
2t


1
1 √ 
− 2
2
1
√

1
−
2
1
√ 
1 √
2
−
2 .
− 2
√
4
1
− 2
1
E1 + E2 + e−i

√
2t
E3
√
U (π/ 2) = −E1 + E2 − E3 .
However


0 0 1

E1 − E2 + E3 = 
0 1 0 ,
1 0 0
which is a permutation matrix and represents the automorphism of P3 that
swaps its end-vertices. It follows that
√ we have perfect state transfer between
the end-vertices of P3 at time π/ 2. It follows that we have perfect state
transfer on the Cartesian powers of P3 .
We can use the spectral decomposition of A to derive a factorization of
U (t). If
X
A=
θr Er ,
r
8
1.7. Bounding Transfer
then the summands commute and therefore
exp(itA) =
Y
exp(itθr Er ).
r
If E is an idempotent, then itE has the spectral decomposition
0 · (I − E) + it · E
and hence
exp(itE) = I − E + eit E.
Thus we have the factorization
UA (t) =
(I − Er + eiθr t Er ).
Y
r
Since I − Er + eiθr t Er acts as the identity on the column space of I − Er and
as multiplication by eitθ (a complex number of norm one) on the column
space of Er , it might be viewed as a complex reflection about the subspace
col(I − Er ). (A complex reflection relative to a subspace U fixes each vector
in U and acts as multiplication by a complex scalar of norm one on U ⊥ .)
1.7
Bounding Transfer
We introduce a useful and interesting bound, and apply it to complete
graphs.
1.7.1 Lemma. If X has spectral decomposition
then
X
|U (t)a,b | ≤
|(Er )a,b |.
P
r
θr Er and a, b ∈ V (X),
r
Proof. We apply the triangle inequality to
U (t) =
X
eitθr Er ,
r
noting that all eigenvalues have norm 1.
9
1. Continuous Quantum Walks
We will study this inequality in greater depth in Section 2.1. For now,
we apply it to the complete graphs.
The eigenvalues of Kn are n − 1 (which is simple) and −1 (with multiplicity n − 1). The corresponding idempotents are
1
J,
n
I−
1
J
n
and consequently
i(n−1)t 1
U (t) = e
n
−it
1
I− J .
n
J +e
Our bound yields that if a and b are distinct vertices in Kn , then
|U (t)a,b | ≤
2
.
n
This implies immediately that the only complete graph with perfect state
transfer is K2 . If we have uniform mixing
on Kn at time t, then each
√
off-diagonal entry of U (t) is equal to 1/ n. However if
1
2
√ ≤
n
n
then n ≤ 4, and so uniform mixing cannot occur on Kn if n > 4. (It is not
hard to show that we do have uniform mixing on K2 , K3 and K4 .)
We note a surprising feature of walks on Kn : if we start the walk in
state v and measure at time t, the probability that the system is in state v
is at least 1 − n2 . Roughly speaking, a continuous quantum walker on Kn
tends to stay at home.
1.8
Bipartite Graphs
The entries of U (t) are complex numbers with norm at most 1. When X is
bipartite though we can say more.
Assume X is bipartite on n vertices and
!
0 B
A=
.
BT 0
10
1.8. Bipartite Graphs
Then
!
2k
A
and
(BB T )k
0
=
,
T
0
(B B)k
A
2k+1
0
(BB T )k B
=
(B T B)k B T
0
!
√
√
!
T)
T )B
BB
i
sin(t
BB
cos(t
√
√
.
U (t) = exp(itA) =
i sin( B T B)B T
cos(t B T B)
Thus we may write
!
C1 (t) iK(t)
U (t) =
.
iK T (t) C2 (t)
where C1 (t) and K(t) are real. Further C1 (t) is an even function and K(t)
is odd.
What we are seeing in the above expression for U (t) is a reflection of
the fact that U (t) is normal and any normal matrix can be written as a
sum C + iS where C and S are commuting Hermitian matrices. (If N is
normal, then C = N + N ∗ and S = −i(N − N ∗ ) are commuting Hermitian
matrices.)
This simplification of the form of U (t) still holds if we allow B to be
an arbitrary real matrix but not if we use the Laplacian, because then the
diagonal is not zero.
The above considerations lead to the following result, due to Kay [7,
Section III].
1.8.1 Lemma. Let X be a bipartite graph. If we have perfect state transfer
from u to v at time t, then the phase factor is ±1 if dist(u, v) is even, and
is ±i if dist(u, v) is odd.
1.8.2 Lemma. Let X be a bipartite graph on n vertices. If uniform mixing
occurs on X then n = 2 or n is divisible by four; if X is regular and uniform
mixing occurs then n is the sum of two squares.
Proof. Assume X is bipartite and U (t) is flat. As we have seen,
√ each
entry of U (t) is either real of purely imaginary, and it follows that nU (t)
must be a complex Hadamard matrix with entries ±1 and ±i. Let D be
a diagonal matrix with Du,u = 1 if u is in the first color class of X, and
Du,u = i otherwise. Then
!
C1 (t) −K(t)
DU (t)D =
K T (t) C2 (t)
∗
11
1. Continuous Quantum Walks
√
and therefore nU (t) is a ±1-matrix. Since it is unitary it is a Hadamard
matrix and therefore n = 2 or n is divisible by four.
If X is regular then all row sums of U (t) are equal and therefore there
are integers a and b such that
√
nU (t)1 = (a + ib)1;
taking complex conjugates yields that
√
nU (−t) = (a − ib)1
and consequently
(a − ib)(a + ib)1 = nU (−t)U (t)1 = n1.
An even cycle is regular and bipartite, so the above result provides
another proof of results of Tamon et al [?]. We have shown that uniform
mixing does not occur on even cycles.
1.9
Vertex-Transitive Graphs
Suppose we have perfect state transfer from u to v in a vertex-transitive
graph X. Assume UX (τ )eu = γev . Then
γP ev = P UX (τ )eu = UX (τ )P eu ,
which means that we have perfect state transfer from the image of u under
P to the image of v. In particular there is a partition of V (X) into pairs
and at time τ we have perfect state transfer between each vertex and its
partner, all with the same phase. Accordingly there is a permutation matrix
T such that
UX (τ ) = γT ;
further all diagonal entries of T are zero and T 2 = I. Since T must commute
with A it is an automorphism of X. Since UX (t) commutes with Aut(X),
it follows that T lies in the center of Aut(X).
1.9.1 Theorem. Assume X is a vertex-transitive graph. If at time τ we
have perfect state transfer from u to v, then there is a complex number
γ and a permutation matrix T such that UX (τ ) = γT . The permutation
matrix T is a fixed-point free automorphism of X with order two that lies
in the center of Aut(X).
12
1.9. Vertex-Transitive Graphs
If X is a Cayley graph for a group G then T must be an element of
G—because the automorphism group of X contains the right regular permutation representation of G and T commutes with each element of this,
so it must belong to the left regular representation of G.
1.9.2 Corollary. If a vertex-transitive graph admits perfect state transfer,
then V (X) is even.
The d-cube Qd is a very relevant example. It is a Cayley graph for Zd2
and admits perfect state transfer from each vertex to its antipode at time
π/2.
Notes
Continuous quantum walks were introduced by Farhi and Gutmann in [5].
The concept of perfect state transfer first appears [1] and the basic theory
was mapped out by Christandl et al. in [2]. In particular the latter paper
shows that perfect state transfer occurs on P2 and P3 , and on their Cartesian
powers. It also offers a proof that, if n > 3, then we do not perfect state
transfer between the end-vertices of Pn . (We will consider this in Section ??,
where we prove that we do not have perfect state transfer between any two
vertices of Pn when n > 3.)
Uniform mixing (or instantaneous uniform mixing) appears first in work
of Moore and Russell [8].
13
Chapter 2
State Transfer
In this chapter we begin a detailed study of perfect state transfer.
2.1
Three Inequalities
We have
U (t)a,b =
X
eitθr (Er )a,b
r
and we get the chain of inequalities
|U (t)a,b | ≤
X
|(Er )a,b |
(2.1.1)
r
≤
Xq
q
(Er )a,a (Er )b,b
(2.1.2)
r
s
≤
X
X
r
r
(Er )a,a
(Er )b,b
(2.1.3)
= 1.
We must explain the individual steps. In (2.1.1) we use the triangle inequality. For (2.1.2), if a and b are vertices in X then by Cauchy-Schwarz
(Er )a,a (Er )b,b − ((Er )a,b )2 = kEr ea k2 kEr eb k2 − hEr ea , Er eb i2 ≥ 0.
Finally
in (2.1.3)qwe use Cauchy-Schwarz again, applied to the vectors
q
( (Er )a,a )r and ( (Er )b,b )r .
We aim to develop a better understanding of these three inequalities,
starting with (2.1.1). Let σr denote the sign of (Er )a,b . (Its value when
(Er )a,b = 0 will be irrelevant, we follow custom and take it to be zero.)
15
2. State Transfer
2.1.1 Lemma. We have |U (t)a,b | ≤ r |(Er )a,b |. Equality holds if and only
if there is a complex number γ such that eitθr = γσr whenever (Er )a,b 6= 0.
If equality holds at time t then U (2t)a,b = 0.
P
Proof. If equality holds in the triangle inequality then the stated condition
must hold. Next,
U (2t)a,b =
X
γ 2 (Er )a,b = γ 2
r
X
(Er )a,b = 0.
r
which completes the proof.
If a ∈ V (X), the numbers (Er )a,a are nonnegative and sum to 1. Hence
they determine a probability density on the eigenvalues of A (whose actual
support is the eigenvalue support of a). We call it the spectral density of
X relative to a. Two vertices have the same spectral density if and only if
they are cospectral. If (pr ) and (qr ) are two probability densities with the
same finite support, their fidelity is defined to be
X√
pr q r .
r
By Cauchy-Schwarz, the fidelity of the densities lies in the interval [0, 1],
and it is equal to 1 if and only if the two densities are equal.
2.1.2 Lemma. The fidelity of the spectral densities of vertices a and b in
P
X is bounded below by r |(Er )a,b |. Equality holds if and only if a and b
are parallel.
Proof. Equality holds in (2.1.2) if and only the vectors Er ea and Er eb are
parallel, for all r.
For (2.1.2), if a and b are vertices in X then by Cauchy-Schwarz
(Er )a,a (Er )b,b − ((Er )a,b )2 = kEr ea k2 kEr eb k2 − hEr ea , Er eb i2 ≥ 0
and equality holds if and only if Er ea and Er eb are parallel.
Finally we equality in (2.1.3) if and only if the spectral densities at a
and b ere equal, equivalently a and b are cospectral.
If we have perfect state transfer from a to b, then |U (t)a,b | = 1, whence
the vertices a and b are parallel and cospectral. Therefore we have another
proof of the following (cf. Lemma ??).
2.1.3 Lemma. If we have perfect state transfer from a to b in the graph
X, then a and b are strongly cospectral.
16
2.2. Symmetry and Periodicity
Note that if (Er )a,b 6= 0 then neither (Er )a,a nor (Er )b,b is zero. So the
eigenvalues θr such that (Er )a,b 6= 0 lie in the intersection of the eigenvalue
supports of a and b.
2.2
Symmetry and Periodicity
We saw that at time π/2 we have perfect state transfer on K2 from vertex
1 to vertex 2 and, at the same time, from vertex 2 to vertex 1. Similarly
on √
P3 we found perfect state transfer for 1 to 3, and from 3 to 1, at time
π/ 2. These are examples of something much more general.
2.2.1 Lemma. If we have perfect state transfer on X from vertex u to
vertex v at time τ , then we have perfect state transfer from v to u at the
same time (and with the same phase factor).
Proof. Suppose U (τ )eu = γev . Then
γ −1 eu = U (−τ )ev
and if we take complex conjugate of both sides, we find
γeu = U (τ )ev .
(Remember that any phase factor has norm 1.)
One consequence of this result is that, if we have uv-pst at time t with
phase factor γ, then
U (2t)eu = U (t)2 eu = γ 2 eu
and, similarly, U (2t)ev = γ 2 ev . We say that X is periodic at u if there is a
time τ such that U (τ )eu = γeu for some γ.
2.2.2 Corollary. If we have perfect state transfer from vertex u to vertex
v in X at time t, then X is periodic at u and v.
Although it is difficult to see any physical applications of periodicity, it
provides a useful mathematical tool for the analysis of perfect state transfer.
We say that a graph X is periodic if there is a time t such that U (t)
is diagonal. Equivalently X is periodic with period t at each vertex. By
virtue of the following lemma, we do not need to make any assumptions on
phase factors.
17
2. State Transfer
2.2.3 Lemma. If X is connected and U (t) is diagonal, then U (t) = γI.
Proof. If X is connected, the only diagonal matrices that commute with A
are the scalar matrices. Since U (t) and commute, the lemma holds.
If the eigenvalues of X are all integers it is easy to verify that X is
periodic. It is a surprising fact that something very close to the converse is
true.
2.3
Periods
The set
Γ = {U (t) : t ∈ R}
is a group, in fact a subgroup of the unitary group. It is abelian and
connected, whence its closure is also abelian and connected. Since
U (t) =
X
eitθr Er ,
r
it follows that we have an isomorphism between Γ and the group formed by
the vectors
(tθ1 , . . . , tθm ),
(t ∈ R)
where the group operation is addition modulo 2π.
Now define Γ(a) to be the set of elements U of Γ such that U ea = γea
for some γ. Clearly Γ(a) ≤ Γ. Next we define the set P (a) of periods at a
by
P (a) = {t : U (t) ∈ Γ(a)}.
2.3.1 Lemma. If a is not an isolated vertex in X and X is periodic at a,
then P (a) consists of all integer multiples of some positive real number τ .
Proof. Set P = P (a). We note that P is an additive subgroup of R and, as
such, is either discrete or dense in R.
If P is dense then there is a sequence of elements (tk )k≥0 of P with limit
0. We have U (tk )ea = γk ea for each k and the limit
1
(U (tk ) − I)ea
k→∞ tk
lim
18
2.3. Periods
exists because U (t) is differentiable and it is equal to
lim
k→∞
γk − 1
ea .
tk
This implies that γk → 0 as k increases. As
1
(U (tk ) − I) = iA
k→∞ tk
lim
we have
lim
1
k→∞ tk
(U (tk ) − I)ea = iAea .
If the vertex a is not isolated, then Aea is not a scalar multiple of ea , and
it follows P is not dense in R. Hence there is a unique positive real number
τ such that P consists of all integral multiples of τ .
We call τ the minimum period of X at a. If we use Q to denote the set of
times at which we have perfect state transfer from a to b, then 2Q ⊆ P , and
therefore if we have perfect state transfer at time t, then 2t is an integral
multiple of τ .
2.3.2 Lemma. Suppose u is a vertex in X with positive valency and X is
periodic at u with minimum period σ. Then if there is perfect state transfer
from u to v, there is perfect state transfer from u to v at time σ/2.
Proof. Suppose we have uv-pst with minimum time τ . Then X is periodic
at u, with minimum period σ (say).
If σ < τ , then U (τ − σ)eu = γev for some γ and so τ is not minimal.
Hence τ < σ. Since X is periodic at u with period 2τ , we see that σ ≤ 2τ .
If σ < 2τ then u is periodic with period dividing 2τ − σ and so σ ≤ 2τ − σ,
which implies that σ ≤ τ . We conclude that σ = 2τ .
Thus if the minimum period of X at u is σ and there is perfect state
transfer from u to v, then there is perfect state transfer from u to v at time
σ/2 (and not at any shorter time).
This result has the following important consequence, first noted by Kay
[7, Section D].
2.3.3 Corollary. If we have perfect state transfer in X from u to v and
from u to w, then v = w.
19
2. State Transfer
It is worth noting that, even though the set of periods P is discrete, the
corresponding phases need not be. If U (t)ea = γea , then γ is an eigenvalue
of U (t), actually
γ = eitθ1
since θ1 belongs to the eigenvalue support of any vertex in a connected
graph. Consequently all phases are powers of eiτ θ1 , and these powers are
dense if and only if τ θ1 /2π is rational. In Section ?? we provide a family
of graphs where this ratio is irrational.
2.4
Bounding the Minimum Period
It is possible to derive a lower bound on the minimum period in terms of
the eigenvalues of X.
2.4.1 Lemma. Suppose X is a graph with eigenvalues θ1 , . . . , θm in decreasing order and transition matrix U (t). If x is a non-zero vector, then
π
.
the minimum time τ such that xT U (τ )x = 0 is at least θ1 −θ
m
Proof. Assume kxk = 1. We want
0 = xT U (t)x =
X
eitθs xT Es x,
where the sum is over the eigenvalues θs such that Er x 6= 0, i.e., over the
eigenvalue support of x. Since
1 = xT x =
X
xT Es x,
the right side is a convex combination of complex numbers of norm 1. When
t = 0 these numbers are all equal to 1, and as t increases they spread out
on the unit circle. If they are contained in an arc of length less than π,
their convex hull cannot contain 0, and for small(ish) values of t, they lie in
the interval bounded by tθ1 and tθm . So for xT UX (t)x to be zero, we need
t(θ1 − θm ) ≥ π, and thus we have the constraint
t≥
π
.
θ1 − θm
If u ∈ V (X) and x = vec u, then this bound is tight for P2 but not for
P3 .
20
2.5. Transition Operators and Walk Modules
2.4.2 Lemma. If X is a graph with eigenvalues θ1 , . . . , θm , the minimum
2π
.
period of X at a vertex is at least θ1 −θ
m
Proof. We want
γ=
X
eiθr t (Er )u,u ,
r
where kγk = 1, and for this to hold there must be integers mr,s such that
t(θr − θs ) = 2mr,s π.
This yields the stated bound.
In the previous lemma, θ1 is the spectral radius of A(X). However θm
can be replaced by the least eigenvalue in the eigenvalue support of the
relevant vertex. If the entries of x are non-negative, these comments apply
to 2.4.1 too. For more bounds along the lines of the last two lemmas, go to
[7, Section IIIC].
For vertex-transitive graphs we can specify the true period: if the eigenvalues are integers and 2d is the largest power of 2 that divides the greatest common divisor of the difference of the eigenvalues, then the period is
π/2d−1 .
√ In Corollary 2.6.3 we will show that the minimum period is at most
2 2π.
2.5
Transition Operators and Walk
Modules
Let a be a vertex in X with dual degree m. Then the vectors Ar ea for
r = 0, . . . , m form a basis for the walk module relative to a. If N is the
matrix with these vectors as its columns then, since col(N ) is A-invariant,
there is a matrix B such that AN = N B. Consequently
U (t)N = N exp(itB).
We can use the follwing to show that in certain cases, the phase factor
γ arising in perfect state transfer is an m-th root of 1, for some m.
2.5.1 Lemma. Suppose X is periodic at the vertex a at time t and with
phase γ. If tr(B) = 0 and the dual degree of a is m, then γ m+1 = 1.
21
2. State Transfer
Proof. If U (t)ea = γea then since U (t) and A commute, U (t)Ar ea = γAr ea
and hence U (t)N = γN . As the columns of N are linearly independent, we
deduce that
exp(itB) = γI.
Now det(exp(itB)) = 1 for all t and det(γI) = γ m+1 .
2.5.2 Lemma. Let B be a square rational matrix. If for some nonzero t all
entries of exp(itB) are algebraic numbers, the ratio of any two eigenvalues
of B is rational.
Proof. Let θ0 , . . . , θd be the distinct eigenvalues of B and set H = exp(itB).
If the entries of H are algebraic, its eigenvalues are algebraic, that is, eitθr
is algebraic for all r. By a version of the Gelfond-Schneider theorem (Theorem ??), if θ1 and θ2 are linearly independent over Q, at least one of the
four numbers
θr , θs , eitθr , eitθs
is transcendental.
2.6
Integrality
Two complex numbers are algebraic conjugates if they are zeros of the same
irreducible monic polynomial, f (t) say, with integer coefficients. If F is a
splitting field for for f (t) then two elements of F are algebraic conjugates if
and only if some element of the Galois group of F maps one to the other. If θ
is an eigenvalue of A, then the entries of idempotent Eθ lie in Q(θ) and hence
if θr and θs are conjugate eigenvalues of A, there is a field automorphism
that maps Er to Es .
Our favorite reference for field theory is Cox [3].
2.6.1 Theorem. Suppose X is connected with at least two vertices and let
S be the eigenvalue support of the vertex u. Then X is periodic at u if and
only if either:
(a) The eigenvalues in S are integers.
√
(b) The eigenvalues in S are all of the form 12 (a + bi ∆), where ∆ is a
square-free integer and a and bi are integers.
22
2.6. Integrality
Proof. It is straightforward to verify that if either of our stated conditions
holds, then X is periodic at u.
So we assume X is periodic at u and seek to show that our conditions
are necessary. As X is connected with at least two vertices, |S| ≥ 2. If θr
and θs are algebraic conjugates, then Er and Es are algebraic conjugates
and so Er eu = 0 if and only if Es eu = 0. Therefore S contains all algebraic
conjugates of each of its elements. If |S| = 2 then either both elements of
S are integers, or they are roots of a quadratic polynomial (and (b) holds).
We assume that |S| ≥ 3.
If two eigenvalues in S are integers, say θ0 and θ1 then since the ratio
conditions asserts that
θr − θ0
∈ Q,
θ1 − θ0
we conclude that all elements of S are integers.
So we may assume at most one element of S is an integer. Let θ0 and θ1
be two distinct elements of S; we will show that (θ1 − θ0 )2 is an integer. By
the ratio condition, if θr , θs ∈ S there is a rational number ar,s such that
θr − θs = ar,s (θ1 − θ0 )
and therefore if δ := |S|, then
Y
(θr − θs ) = (θ1 − θ0 )δ
2 −δ
r6=s
Y
ar,s .
i6=j
The product on the left is an integer and the product of the ar,s ’s is rational,
and hence
2
(θ1 − θ0 )δ −δ ∈ Q.
Since θ1 − θ0 is an algebraic integer, this implies that
(θ1 − θ0 )δ
2 −δ
∈ Z.
Suppose m is the least positive integer such that (θ1 − θ0 )m is an integer.
Then there are m distinct conjugates of θ1 − θ0 of the form
βe2πik/m
(k = 0, . . . , m − 1)
where β is the positive real m-th root of an integer, and since the eigenvalues
of X are real we conclude that m ≤ 2. Therefore θ1 − θ0 is either an integer
or the square root of an integer. Since
(θr − θs )2 = a2r,s (θ1 − θ0 )2
23
2. State Transfer
it follows that (θr − θs )2 is rational and therefore it is an integer. So θr − θs
is an integer multiple of the square root of an integer ∆r,s
If θk and θ` are elements of S we have
(θk − θ` ) = ak,` (θ1 − θ0 )
and consequently
(θr − θs )(θk − θ` ) = ar,s ak,` (θ1 − θ0 )2 .
Therefore (θr − θs )(θk − θ` ) is an integer. This implies that ∆k,` = ∆r,s , and
hence we can write
√
θr = θ0 − mr ∆
where ∆ is a square free integer and mr is an integer. The spectral radius
lies in S, so we assume it is θ0 and it follows that the integers mr are
positive.
We now
√ distinguish two cases. Suppose first that θ0 is an integer. Then
θ0 − m1 ∆ ∈ S and,
√ since S contains all algebraic conjugates of each of its
elements, θ0 + m1 ∆ ∈ S, which is a contradiction. So we may assume θ0
is irrational. The sum of the elements of S is an integer and therefore
√ X
mr ∈ Z.
|S|θ0 − ∆
r
√
Therefore θ0 ∈ Q( ∆) and, as it is an algebraic integer it follows (from [6,
p. 54] for example) that we can write it as
√
1
(a + b ∆).
2
2.6.2 Corollary. There are only finitely many connected graphs with maximum valency at most k where perfect state transfer occurs.
Proof. Suppose X is a connected graph where perfect state transfer from
u to v occurs at time τ and let S be the eigenvalue support of u. If the
eigenvalues in S are √
integers then |S| ≤ 2k + 1, and if they are not integers
then |S| < (2k + 1) 2. So the dimension of
√ the A-invariant subspace of
V (X)
R
generated by eu is at most d(2k + 1) 2e, and this is also a bound
on the maximum distance from u of a vertex in X. If s ≥ 1, the number of
vertices at distance s from u is at most k(k − 1)s−1 , and the result follows.
24
2.6. Integrality
2.6.3 Corollary. If X is periodic
at a vertex with period τ and τ̂ := τ /2π,
√
then τ̂ 2 is rational and τ̂ ≤ 2.
Proof. Suppose X is periodic at u and let S be the eigenvalue support of u.
If the elements of S are integers then
U (2π)u,u =
X
θr ∈S
e2πiθr (Er )u,u =
X
(Er )u,u = 1.
r
√
Otherwise the elements of S are of the form (a + bi ∆)/2 and then
√
√
√
X
e4πia/ ∆ e2πibr (Er )u,u = e4πia/ ∆ .
U (4π/ ∆)u,u =
r
To show that τ̂ 2 is rational we first observe that, since U (τ )u,u = γ we
have
eiθr τ = eiθs τ
for any eigenvalues θr and θs in the eigenvalue support of u. Hence there is
an integer mr,s such that
τ (θr − θs ) = 2mr,s π.
Since (θr − θs )2 is an integer, it follows that τ̂ 2 is rational.
2.6.4 Lemma. If u ∈ V (X) and the minimal polynomial of A(X) relative
to eu is irreducible, then u is not involved in perfect state transfer.
Proof. Suppose we have perfect state transfer from u to v and let N be
the A-module generated by eu . Then N is the direct sum of the A-modules
generated by eu −ev and eu +ev , neither of which is zero. Hence the minimal
polynomial of A restricted to U factorizes non-trivially.
2.6.5 Corollary. If φ(X, t) is irreducible there is no perfect state transfer
on X.
Proof. The minimal polynomial of A(X) relative to eu divides φ(X, t).
If u and v are cospectral and ev belongs to the A-module generated by
u, then the A-modules generated by eu and ev are equal, because they have
the same dimension.
25
2. State Transfer
2.7
No Control
We will prove that a controllable vertex cannot be involved in perfect state
transfer. For this we need the following:
2.7.1 Lemma. Let X be a graph with n = |V (X)|. If σ is the minimum
distance between two eigenvalues of X, then
σ2 <
12
.
n+1
Proof. Assume that the eigenvalues of X in non-increasing order are θ1 , . . . , θn .
If we have
M := A ⊗ I − I ⊗ A
then the eigenvalues of M are the numbers
θi − θj ,
1 ≤ i, j ≤ n.
Now
M 2 = A2 ⊗ I + I ⊗ A2 − 2A ⊗ A
and consequently, if e := |E(X)| then
n
X
(θi − θj )2 = tr(M 2 ) = 2n tr(A2 ) = 4ne.
i,j=1
Since
θi − θj ≥ (i − j)σ
we have
n
X
(θi − θj )2 ≥ σ 2
i,j=1
As
n
X
n
X
(i − j)2
i,j=1
i2 =
i=1
n(n + 1)(2n + 1)
6
we find that
n
X
(i − j) = 2n
i,j=1
26
2
n
X
i=1
2
i −2
n
X
i=1
!2
i
=
n2 (n2 − 1)
,
6
2.7. No Control
and since e ≤ n(n − 1)/2 this yields
σ2
n2 (n2 − 1)
≤ 4ne ≤ 2n2 (n − 1).
6
This gives our stated bound but with ≤ in place of <. To achieve strictness
we note that if equality were to hold then e = n(n − 1) and X = Kn . Since
σ(Kn ) = 0, we are done.
2.7.2 Theorem. Let X be a connected graph on at least four vertices. If
we have perfect state transfer between distinct vertices u and v in X, then
neither u nor v is controllable.
Proof. Let n = |V (X)|. We assume by way of contradiction that (X, u) is
controllable. Then the eigenvalue support of u contains all eigenvalues of
X and these eigenvalues are distinct. By 2.6.1, this means that there and
integer ∆ and distinct integers b1 , . . . , bn such that the eigenvalues of A are
the numbers
√
br ∆.
It follows that the separation σ(X) between distinct eigenvalues is at least
1.
By 2.7.1 we can assume that n = |V (X) ≤ 10. This leaves us with six
cases. First suppose ∆ = 1. Assume n = 10. Then the sum of the squares
of the eigenvalues of X is bounded below by the sum of the squares of the
integers from −4 to 5, which is 85, and hence the average valency of a vertex
is at least 8.5. This implies that θ1 = 9, not 5, consequently the sum of the
squares of the eigenvalues is at least
85 − 25 + 81 = 141
and now the average valency is 14.1, which is impossible. The cases where
n is 7, 8 or 9 all yield contradictions in the same way. If ∆ > 1, it is even
easier to derive contradictions.
Next, brute force computation (using Sage [9]) shows that the path P4 is
the only graph on 4, 5 or 6 vertices where the minimum separation between
consecutive eigenvalues is at least 1. The positive eigenvalues of P4 are
√
( 5 ± 1)/2
and their ratio is not rational.
27
Chapter 3
Real Lines
The first definition is easy enough: a set of lines in Rn is equiangular if
the angle between any two distinct lines is the same. The simplest example
would be the coordinate axes in Rd , which gives us a set of size d. The first
problem is to determine the maximum size of a set of equiangular lines in
Rd .
3.1
Projections
A line ` is a 1-dimensional subspace of V and hence can be represented by
a basis, that is, any non-zero vector in `. We can reduce redundancy by
choosing a unit vector as a basis but even over the reals this still leaves a
choice—between −x and x—while over C a unit vector is only determined
up to multiplication by a complex number of norm 1. (In quantum physics
these are known as phase factors.)
We can eliminate redundancy by using projections. If U is a subspace
of V with an orthonormal basis u1 , . . . , ud then the matrix
PU :=
d
X
ui u∗i
i=1
represents orthogonal projection onto. Hence
P = P∗ = P2
and U is the image of PU and U ⊥ is its kernel. In particular
tr(PU ) = rk(PU ) = dim(U ).
29
3. Real Lines
The space of linear operators on V is an inner product space, with inner
product
hA, Bi := tr(A∗ B) = sum(A ◦ B).
If P and Q are projections then
kP − Qk2 = tr((P − Q)2 )
= tr(P 2 + Q2 − P Q − QP )
= tr(P ) + tr(Q) − 2 tr(P Q)
= tr(P ) + tr(Q) − 2hP, Qi.
If P and Q are projections onto subspaces of dimension d, it follows that
kP − Qk2 = 2d − hP, Qi;
if P and Q are projections onto lines spanned by unit vectors u and v
respectively then
P = uu∗ , Q = vv ∗
and
tr(P Q) = tr(uu∗ vv ∗ ) = tr(v ∗ uu∗ v) = |u∗ v|2
whence
kP − Qk2 = 2 − 2|u∗ v|2 .
If we are working over R then |u∗ v| is the cosine of the angle between the
lines spanned by u and v. Hence we will call |u∗ v|2 a squared cosine, even
over C.
Projections are Hermitian matrices, and the Hermitian matrices of order
d × d form a real vector space of dimension d2 . Over R our projections are
real symmetric matrices, and these form a real vector space of dimension
d+1
.
2
3.2
Equiangular Lines
We begin by a deriving sharp upper bound on the size of a set of equiangular
lines in Rd . The first step is another representation of lines. Suppose x is a
non-zero vector. Then the matrix
X=
30
1
xxT
hx, xi
3.2. Equiangular Lines
is symmetric and idempotent and its image is the line spanned by x. Thus
X represents orthogonal projection onto the line spanned by x. Note that if
we replace x by c, where c 6= 0, the matrix X does not change. In particular
x and −x give rise to the same matrix X. Thus X represents our line and
does not depend on the choice the basis of the line.
Further, suppose xi and xj are unit vectors and
Xj = xj xTj .
Xi := xi xTi ,
Then
Xi Xj = hxi , xj ixi xTj
and
hXi , Xj i = tr(Xi Xj ) = hxi , xj i2 .
Thus hXi , Xj i is the squared cosine of the angle between the lines spanned
by xi and xj . Also hXi , Xi i = 1.
3.2.1 Theorem. A set of equiangular lines in Rd has size at most
d+1
2
.
Proof. Suppose our lines are spanned by vectors x1 , . . . , xn , with corresponding projections X1 , . . . , Xn , and that the square cosine is γ. We prove that
the matrices X1 , . . . , Xn are linearly independent. Since these matrices lie
in the real vector space of d × d symmetric matrices, which has dimension
!
d+1
,
2
the theorem follows immediately.
Assume that c1 , . . . , cn are scalars and
0=
n
X
ci X i .
i=1
Take the inner product of each side with Xr . Then
0 = cr +
X
ci γ
i6=r
= (1 − γ)cr + γ
X
ci .
i
31
3. Real Lines
Since this holds for r = 1, . . . , n, we se that cr is independent of r. Therefore
we must have
n
0=
X
Xi ,
i=1
but the trace of the right side is n, and so it cannot be zero. We conclude
that the matrices Xi are linearly independent.
The above bound on the size of an equiangular set of lines is known as
the absolute bound. You may convince yourself that it is tight in R2 . (We
will consider the question of tightness in more detail later.) Note that if we
have an equiangular set of n lines in Rd , the intersection of these lines gives
us a set of 2n points, namely the 2n unit vectors that span the lines. If xi
and xj are two of these vectors, then
√
kxi − xj k2 = 2 − 2hxi , xj i ≥ 2 − 2 γ
Using this and some spherical geometry, we could derive an upper bound
on the size of our set. However the resulting bound depends on γ and is
exponential in d.
3.3
The Relative Bound
We derive a second bound on the size of an equiangular set of lines. This
bound depends both on d and the squared cosine γ. (It will also be easier
to give examples where it is tight.)
Suppose X1 , . . . , Xn are the projections onto a set of equiangular lines in
Rd with squared cosine γ. If the number of lines meets the absolute bound,
then these projections span the vector space of d × d symmetric matrices,
whence there are scalars ci such that
I=
X
ci X i .
i
If we take the inner product of each side with Xr , we find that
1 = (1 − γ)cr + γ
X
ci
i
As before this implies that cr is independent of r. Comparing traces, we
conclude that, if the absolute bound holds, then
X
n
Xi = I.
d
i
32
3.3. The Relative Bound
This may motivate the following. We forget the absolute bound and
compute
νI −
X
i
Xi , νI −
X
Xi = ν 2 d − 2νn + n + (n2 − n)γ
i
Here the left side is nonnegative for any choice of ν, so we substitute n/d
for ν in the right side and deduce that
n2
n2
− 2 + n + (n2 − n)γ
0≤
d
d
n
= (−n + d + d(n − 1)γ)
d
and consequently
n(1 − dγ) ≤ d − dγ.
If dγ < 1, we conclude that the following relative bound holds.
3.3.1 Theorem. If there is an equiangular set of n lines in Rd with squared
cosine γ, then
d − dγ
n≤
1 − dγ
and, if equality holds and X1 , . . . , Xn are the projections onto the lines,
then
X
n
Xi = I.
d
i
If equality holds, we will see that the possible values for γ are quite
restricted. If
X
n
Xi =
d
i
then taking the inner product of each side with X1 yields
1 − γ + nγ =
n
d
and consequently
γ=
n−d
.
d(n − 1)
33
3. Real Lines
In particular, if we have an equiangular set of
projections X1 , . . . , Xn , then
γ=
d+1
2
lines in Rd with
1
.
d+2
A set of unit vectors x1 , . . . , xn forms a tight frame in Rd if
X
xi xTi =
i
3.4
n
I.
d
Gram Matrices
Suppose x1 , . . . , xn is a set of unit vectors in Rd that span a set of equiangular lines with squared cosine γ, and let G be their Gram matrix. Then
we may write
√
G = I + γS,
where S is a symmetric matrix with all diagonal entries zero, and all offdiagonal entries equal to ±1. We call S the Seidel matrix of the set of
vectors. Further
1
(J − I + S)
2
is the adjacency matrix of a graph. Since the lines are determined up
to an orthogonal transformation by the gram matrix, it follows that each
equiangular set of lines is determined by a graph. (The correspondence is
many-to-one, since we may replace xi by −xi without changing the set of
lines. The leads to the concept of switching classes of graphs, but we do
not go into this now.)
The next result will be the key to our analysis.
3.4.1 Lemma. Suppose x1 , . . . , xn is a set of unit vectors in Rd that span
a set of equiangular lines with squared cosine γ and let G be their Gram
matrix. If the relative bound holds with equality, then
G2 =
n
G.
d
Proof. Let Xi = xi xTi . If the relative bound is tight, then
X
i
34
Xi =
n
I.
d
3.4. Gram Matrices
Let U be the d × n matrix with x1 , . . . , xn as its columns. Then
X
Xi = U U T
i
and G = U T U . Hence
G2 = U T (U U T )U =
n
n T
U U = G.
d
d
It follows that the minimal polynomial of G is
t2 −
n
t
d
and therefore the eigenvalues of G are n/d (with multiplicity d) and 0 (with
multiplicity n − d). If
1
S = √ (G − I)
γ
then the eigenvalues of S are
1
n−d
√ , −√
d γ
γ
with respective multiplicities d and n − d. A symmetric matrix with one
eigenvalue is a scalar multiple of I; we have found that if there is an equiangular set of lines meeting the relative bound, then the Seidel matrix S is a
symmetric matrix with only two eigenvalues. Note that the procedure is reversible: given a Seidel matrix with only two eigenvalues, we can construct
a set of equiangular lines with size meeting the relative bound.
If S is a Seidel matrix with exactly two eigenvalues α and β, then
0 = (S − αI)(S − βI) = S 2 − (α + β)S + αβI.
Since the diagonal of S is zero, it follows that each diagonal entries of S 2
is equal to −αβ. On the other hand since the off-diagonal entries of S are
all ±, each each diagonal entry of S 2 is equal to n − 1. Thus we see that
the product of the eigenvalues of S is 1 − n. Hence the eigenvalues of S are
√
√
(n − 1) γ and −1/ γ.
35
3. Real Lines
3.5
Number Theory
Suppose F and E are fields and F ≤ E. If a ∈ E, the minimum polynomial
of a over F is the monic polynomial ψ of least degree with coefficients in F
such that ψ(a) = 0, if it exists. We will only be concerned with cases where
F = Q and E = C. An element which does not have a minimal polynomial is
transendental, otherwise it is algebraic. The elements of C whose minimal
polynomial over Q have integer coefficients are called algebraic integers;
these form a ring. The minimal polynomial of a over F is irreducible over
F. Two elements of E are algebraic conjugates over F if they have the same
minimal polynomial.
Note that E is a vector space over F, we denote its dimension by |E : F|
and, if it is finite, we may say that E is an extension of F with degree equal
to E : F|. A quadratic extension is an extension of degree two. If a ∈ E,
then the set F[a] of all polynomials in a with coefficients from F is a vector
space over F; its dimension is the degree of the minimal polynomial of a.
Suppose φ(t) is the characteristic polynomial of the integer matrix A.
If λ is an eigenvalue of A, thenφ(λ) = 0 and it follows that the minimal
polynomial of a divides φ. Hence each zero of the minimal polynomial, that
is, each algebraic conjugate of λ, is an eigenvalue of A. Further all algebraic
conjugates of λ will have the same algebraic multiplicity.
3.5.1 Lemma. If there is an equiangular set of n lines in Rd with squared
√
cosine γ such that the relative bound holds, then either 1/ γ is an integer,
√
or n = 2d and 1/ γ lies in a quadratic extension of the rationals.
Proof. Since S is an integer matrix, its eigenvalues are algebraic integers.
Further if λ is an eigenvalue of S, then all its algebraic conjugates are
eigenvalues of S with multiplicities equal to the multiplicity of λ. Since S
has exactly two eigenvalues with multiplicities n−d and d we see that either
√
λ is an integer, or n = 2d and λ is a quadratic irrational. Since −1/ γ is
an eigenvalue of S, the second claim follows.
We have seen that if we have an equiangular set of
then γ = (d + 2)−1 . Hence we have the following.
d+1
2
lines in Rd ,
3.5.2 Corollary. If there is an equiangular set of lines in Rd meeting the
absolute bound and d ≥ 4, then d + 2 is the square of an integer.
36
3.6. Switching
We will see later that d+2 must actually be the square of an odd integer.
Examples of sets of lines meeting the absolute bound are known when d = 2,
3, 7 or 23 (and we will present them later). No other examples are known.
3.6
Switching
If X is a graph then
S = A(X) − A(X)
is a Seidel matrix and so, if X has v vertices and least eigenvalue τ with
multiplicity m, then
S − γI
is the Gram matrix of an equiangular set of v lines in Rv−m with squared
cosine γ −2 . Suppose D is a v × v diagonal matrix with diagonal entries
±1. Then D = D−1 and so DSD is a Seidel matrix which is similar to
S. As far as lines are concerned, replacing S by DSD is equivalent to
multiplying some of the spanning unit vectors −1, and so geometrically
nothing interesting is happening. However DSD is the Seidel matrix of
some graph Y , and we want to determine the relation between X and Y .
If σ ⊆ V (X), we define X σ to be the graph we get from X by complementing the edges that join vertices in σ to vertices not in σ. If σ denotes
the complement of σ, then in set theoretic terms E(X σ ) is the symmetric
difference of E(X) and the edge set of the complete bipartite graph with
bipartition
(σ, σ)
We say that X σ is obtained by switching about the subset σ. Note that
switching twice about σ restores X to itself, and that
Xσ = Xσ.
If D is the v × v diagonal matrix such that Di,i = −1 if i ∈ σ and Di,i = 1
if i ∈
/ σ, then
DS(X)D = S(X σ ).
This reconciles the graph theory and the linear algebra.
It is not hard to show that any sequence of switchings on subsets of
X can be realised by switching on a single subset. So we say graphs X
and Y are switching equivalent if Y is isomorphic to X σ for some σ, and
37
3. Real Lines
the graphs that are switching equivalent to X form its switching class. If
σ is the neighborhood of a vertex v in X, then v is an isolated vertex in
X σ ; in this case we say that X σ is obtained by switching off v. In 3.10
we introduce the graph of a set of equiangular lines; this determines the
switching class of X. (More precisely, it reduces switching equivalence of
graphs on v vertices to isomorphism of certain graphs on 2v vertices.)
3.7
Paley Graphs
Let F be a finite field of order q, where q ≡ 1 modulo four. The Paley graph
on q vertices has F as its vertex set, and two field elements are adjacent if
and only if their difference is a non-zero square in F. (The condition on q
assures that we obtain a graph rather than a directed graph. The 5-cycle
is the Paley graph associated to the field of order 5.
A Paley graph is self-complementary and is strongly regular with parameters
q−1 q−5 q−1
;
,
.
q,
2
4
4
Its eigenvalues are its valency (with multiplicity 1) and
1
√
(1 ± q).
2
each with multiplicity (q − 1)/2.
3.7.1 Lemma. If X is a Paley graph on q vertices and S = S(X ∪ K1 ),
then S 2 = qI.
Proof. Exercise.
√
Since tr(S) = 0, the eigenvalues ± q each have multiplicity (q + 1)/2.
Hence
√
S + qI
is the Gram matrix of a equiangular set of q + 1 lines in R(q+1)/2 .
In particular, the Paley graph on five vertices provides us with a set of
six equiangular lines in R3 . This realizes the absolute bound (and provides
a construction of the icosahedron).
A v × v matrix C with zero diagonal and entries ±1 off the diagonal is
a conference matrix if
C T C = (v − 1)I.
38
3.8. A Spherical 2-Design
The Seidel matrices we have just constructed are symmetric conference
matrices.
3.8
A Spherical 2-Design
Suppose we have a set of n equiangular lines in Rd with squared cosine γ.
We may assume without loss that one of the lines is spanned by the first
standard basis vector e1 , and then we can choose unit vectors x2 , . . . , xn
spanning the remaining n − 1 lines so that
he1 , xi i =
√
γ.
This means that each vector xi can be written as
√ !
γ
xi =
yi
where kyi k =
the form
√
1 − γ. Hence the projection onto the line spanned by xi has
√
!
γyiT
.
yi yiT
γ
√
γyi
Now assume that the relative bound is tight. If X1 , . . . , Xn denote the
projections onto our lines, then
n
X
Xi =
i=1
n
I.
d
If we let zi denote the unit vector (1 − γ)−1/2 yi , then we have
X
i
zi = 0,
X
i
zi ziT =
n
I.
d − dγ
It follows that the vectors zi provide an example of what is known as a a
spherical 2-design. For now we simply show that these vectors determine a
strongly regular graph on n − 1 vertices.
The first step is to note that since
√
γ + yiT yj = xTi xj = ± γ
39
3. Real Lines
we have
√
± γ−γ
.
=
1−γ
We define a graph G with the vectors zi as its vertices, where two distinct
vectors are adjacent if their inner product is positive. Let Z denote the
(d − 1) × (n − 1) matrix with the vectors zi as its columns. If A := A(G),
then
√
√
γ−γ
γ+γ
T
A−
(J − I − A)
Z Z=I+
1−γ
1−γ
√
√
√
1+ γ
2 γ
γ+γ
=
I+
A−
J.
(3.8.1)
1−γ
1−γ
1−γ
ziT zj
On the other hand
ZZ T =
X
zi ziT =
i
n
I
d − dγ
and therefore
n
Z T Z.
d − dγ
T
This implies that Z Z has exactly two eigenvalues, and from (3.8.1) it
P
follows that A has exactly three eigenvalues. Since zi = 0, we see that
(Z T Z)2 =
JZ T Z = Z T ZJ = 0,
and therefore G is regular. Thus G is a regular graph with three eigenvalues,
and therefore it is strongly regular. (A strongly regular graph can arise in
this way if and only if k = 2c.)
You are invited to show that if z is an eigenvector of A that is orthogonal
to 1, then its eigenvalue is one of
1
2
!
n−d
1
√
√ − 1 = [(n − 1) γ − 1],
d γ
2
!
1
1
−√ − 1 .
2
γ
If we compare these with the eigenvalues of the Seidel matrix, we deduce
that if the eigenvalues of the Seidel matrix are integers, they must be odd
integers.
For any strongly regular graph, c − k = θτ and as k = 2c it follows that
k = −2θτ . Hence the valency of G is
1
√
√
k = √ ((n − 1) γ − 1))(1 + γ).
2 γ
40
3.9. An Example
3.9
An Example
We construct a set of 28 vectors xi,j in R8 by defining xi,j to be the vector
with i-th and j-th entries equal to 3, and all other entries equal to −1. The
entries of each of these vectors sum to zero, and so they span a set of lines
in R7 with squared cosine 1/9. Since
!
8
28 =
,
2
we have equality in the absolute bound. Choose x1 to be the vector with
first two entries equal to 3 (which is not a unit vector but that will not
matter). The neighbors of x1 in the graph of the lines consists of the 27
vectors of the form ±xi,j with positive inner product with x1 . This set of
vectors consists of the 12 vectors with first or second entry equal to 3, and
the fifteen vectors with first two entries equal to 1.
The eigenvalues of the Seidel matrix are 9 and −3, and the eigenvalues
of the neighborhood in the two-graph are 4 and −2. The valency is 16. If
the eigenvalues of the neighborhood are k, θ and τ , then
(t − θ)(t − τ ) = t2 − (a − c)t − (k − c).
Hence we have
c = k + θτ,
a = c + θ + τ = k + θτ + θ + τ
and for the graph at hand
c = 8,
3.10
a = 10.
Graphs from Equiangular Lines
Let x1 , . . . , xn be a set of n unit vectors in Rd , spanning a set of equiangular
lines with squared cosine γ. The graph of this set of lines has the 2n vectors
±xi as its vertices, and two such vectors are deemed to be adjacent if their
√
inner product is γ. Thus its vertex set is partitioned into n pairs
{xi , −xi }
and if j 6= i then xj is adjacent to exactly one of the vectors xi and −xi .
So the subgraph induced by two pairs is isomorphic to 2K2 .
41
3. Real Lines
3.10.1 Lemma. If Y is the graph of a set of equiangular lines, then we
may write A(Y ) in the form
!
A(X) A(X)
A(Y ) =
A(X) A(X)
where A(X) − A(X) is the Seidel matrix of the set of lines.
3.10.2 Corollary. Suppose Y is the graph of a set of n equiangular lines.
Then
φ(A(Y ), t) = φ(S, t)φ(Kn , t).
Proof. If
!
A A
A(Y ) =
A A
where A = A(X) for some graph X, then
I 0
I I
!
!
A A
A A
!
!
A
I 0
A−A
=
.
−I I
0
A+A
As an exercise, you may prove that if the graph of an equiangular set of
n lines is either connected with diameter three, or is isomorphic to 2Kn .
If u, v and w are distinct vertices in the graph of an equiangular set of
lines and
dist(u, v) = dist(u, w) = 3
then
N (v) \ w = N (w) \ c,
since N (v) and N (w) are sets of size n − 1 disjoint from u ∪ N (u).
3.10.3 Theorem. If L is an equiangular set of lines in Rd that meets the
relative bound, then its graph is an antipodal distance-regular graph of
diameter three, and the neighbourhood of any vertex is strongly regular.
Proof. We have already proved the second claim, in 3.8. Given this it is
easy to show that the graph is distance regular and antipodal with diameter
three. Do it.
42
3.11. Distance-Regular Graphs
The graph of a set of equiangular lines is sometimes called a two-graph;
we say a two-graph is regular if each neighborhood is regular. A two-graph
is regular if and only if the size of corresponding set of lines meets the
relative bound.
3.10.4 Theorem. If Y is a two-graph, the following are equivalent:
(a) Y is distance regular.
(b) The neighborhood of each vertex of Y is regular.
(c) The neighborhood of each vertex of Y is strongly regular.
(d) The neighborhood of some vertex is strongly regular with k = 2c.
3.11
Distance-Regular Graphs
If X is a graph define its i-th adjacency matrix Ai to be the symmetric
01-matrix with rows and columns indexed by V (X) and with
(Ai )u,v = 1
if and only the distance in X between u and v is i. Thus A0 = I and
X
Ai = J.
i
We may take Ai to be undefined or zero if i is greater than the diameter d
of X. We will also refer to the matrices Ai as the distance matrices of X.
If i ≥ 1, then Ai is the adjacency matrix of a graph with vertex set V (X);
we call it the i-th distance graph of X.
We say a graph X is distance regular if given integers i and j and
vertices u and v, the number of vertices x in X at distance i from u and
distance j from v is determined by i, j and the distance between u and j.
If X is distance-regular and dist(u, v) = r, then we denote the number of
vertices x in X at distance i from u and distance j from v by pi,j (r) and
we call it an intersection number. Cycles and complete graphs are distance
regular, as are the complete bipartite graphs Kn,n . A distance-regular graph
of diameter two is a connected strongly-regular graph.
Our first result translates the property of being distance regular into
matrix terms.
43
3. Real Lines
3.11.1 Lemma. A graph X with diameter d is distance-regular with intersection numbers pi,j (r) if and only if
Ai Aj =
d
X
pi,j (r)Ar .
i=0
The intersection numbers pi,j (k) are not independent, for example
pi,j (r) = pj,i (r).
(Note that this implies that Ai Aj = Aj Ai for all i and j.) Also if |i − r| ≥ 2,
we have
pi,1 (r) = 0
We adopt the convention that Ai = 0 if i < 1 or i > d.
3.11.2 Theorem. A graph X with diameter d is distance-regular if and
only if there are non-negative integers ai , bi , ci such that for i = 0, . . . , d,
Ai A1 = ci Ai−1 + ai Ai + bi Ai+1 .
Proof. The given conditions are a subset of the conditions for distanceregularity, so they are certainly necessary.
We prove they are sufficient. Since X has diameter d, the numbers
b1 , . . . , bd−1
are positive, and so these conditions imply that, for i = 1, . . . , d, there
is a polynomial fi such that Ai = fi (A1 ) and, moreover, that the vector
space spanned by A0 , . . . , Ai is equal to the space spanned by I, A, . . . , Ai .
Therefore the distance matrices A0 , . . . , Ad form a basis for the space of
all polynomials in A1 , and so Ai Aj is a linear combination of distance
matrices.
A graph X with diameter d is antipodal if Xd is a disjoint union of
cliques. A distance-regular graph is imprimitive if Xi is not connected, for
some i between 1 and d. If X is bipartite then X2 is not connected.)
3.11.3 Theorem. Let X be a distance-regular graph with diameter d. If
i ≥ 1 and Xi is not connected then either i = 2 (and X is bipartite) or
i = d (and X is antipodal).
44
3.12. Strongly Regular Graphs
The 3-cube is distance-regular, bipartite and antipodal (and so are the
even cycles). The line graph of the Petersen graph is distance-regular and
antipodal but not bipartite. The intersection matrix of the 4-cube is

0

1

0


0
0
4
0
2
0
0
0
3
0
3
0
0
0
2
0
4

0

0

0

1

0
A graph X is distance transitive if, given two pairs of vertices (x1 , x2 )
and (y1 , y2 ) such that
dist(x1 , x2 ) = dist(y1 , y2 ),
there is an automorphism, g say of X such that
xg1 = y1 ,
xg2 = y2 .
A distance-transitive graph is necessarily distance regular, and verifying
that a graph is distance-transitive may be the easiest way of verifying that
it is distance regular. Note though that there are many distance-regular
graphs which are not distance transitive.
3.12
Strongly Regular Graphs
A graph X is strongly regular if there are integers k, a and c such that
(a) X is k-regular, and is not complete or empty.
(b) Two adjacent vertices have exactly k common neighbours.
(c) Two distinct non-adjacent vertices have exactly c common neighbours.
If X is strongly regular with v vertices, we call it a (v, k, a, c) strongly
regular graph.
Show that if X is strongly regular, then so is its complement, and express
its parameters in terms of the parameters of X. The line graphs of the
complete graphs and the line graphs of the complete bipartite graphs are
strongly regular.
45
3. Real Lines
A connected graph is strongly regular if and only if it is distance regular
with diameter two. If m, n > 1 then mKn is strongly regular but is not
distance regular. A strongly regular graph X is primitive if X and X
are both connected. The only imprimitive strongly regular graphs are the
graphs mKn (where m, n > 1) and their complements.
If X is strongly regular with parameters (v, k, a, c), then
A2 = kI + aA + c(J − I − A)
and consequently,
A2 − (a − c)A − (k − c)I = cJ.
Conversely, if there is a quadratic polynomial f (t) such that f (A) = J, then
X is strongly regular.
The eigenvalues of a strongly regular graph with parameters (v, k, a, c)
are its valency k and the two zeros of the polynomial
t2 − (a − c)t − (k − c);
we denote these zeros by θ and τ , and usually work with the convention
that τ < θ. (In practice this means that τ < 0 < θ.)
The intersection matrix of a strongly regular graph with parameters
(v, k, a, c) is


0 k
0


1 a k − a − 1 .
0 c
k−c
The characteristic polynomial of this matrix is
(t − k)(t2 − (a − c)t − (k − c)).
3.13
SRG’s: Examples
The line graph of the complete graph Kn is a strongly regular graph with
parameters
!
!
n
, 2n − 4, n − 2, 4 .
2
Its eigenvalues are its valency 2n − 4 and the zeros of
t2 − (n − 6)t − (2n − 8) = (t + 2)(t − n + 4).
46
3.14. Equitable Partitions
Thus θ = n − 4 and τ = −2. The matrix of eigenvalues is


1 2n − 4 n−2
2 

1 n − 4 3 − n  .


1
−2
1
and if we denote the multiplicities of n − 4 and −2 by f and g respectively
then
1
f + g = (n2 − n) − 1
2
f (n − 4) − 2g = −2n + 4.
Hence f = n − 1 and g = 21 (n2 − 3n).
The line graph of the complete bipartite graph Kn,n is strongly regular
with parameters (n2 , 2n − 2, n − 2, 2) and hence its eigenvalues are the
valency 2n − 2 and the zeros of
t2 − (n − 4)t − (2n − 4) = (t + 2)(t − n + 2).
The matrix of eigenvalues is
1 2n − 2 (n − 1)2

1−n 
1 n − 2

1
−2
1


and the respective multiplicities of n − 2 and −2 are 2n − 2 and (n − 1)2 .
The Schläfli graph is a strongly regular graph with parameters (27, 10, 1, 5).
So its eigenvalues are 10 and the zeros of
t2 + 4t − 5 = (t + 5)(t − 1).
Thus θ = 1 and τ = −5 and their respective multiplicities are 20 and 6.
(The vertices of the Schläfli graph were originally the 27 lines on a smooth
cubic surface, where two lines were adjacent if they had a point in common.
We will offer another definition later.)
3.14
Equitable Partitions
We introduce an important tool for getting information about eigenvalues
and eigenvectors for graphs.
47
3. Real Lines
3.14.1 Lemma. Suppose U is an n × m matrix and A is n × n. Then
col(U ) is A-invariant if and only if there is an m × m matrix B such that
AU = U B.
3.14.2 Lemma. If AU = U B and the columns of U are linearly independent, then the characteristic polynomial of B divides the characteristic
polynomial of A.
Proof. Let u1 , . . . , um denote the columns of U ; since these vectors are
linearly independent, there are vectors um+1 , . . . , un that together with
u1 , . . . , um form a basis for Rn . The matrix representing A relative to this
basis has the form
!
B C
0 D
where D is square. The characteristic polynomial of this matrix is equal to
φ(B, t)φ(D, t)
and so the result follows.
We will construct A-invariant subspaces using partitions. Suppose π is
a partition of the set V . We use |π| to denote the number of cells of π. We
can represent a partition by its characteristic matrix which is the 01-matrix
of order |V | × |π| whose i-th column is the characteristic vectors of the i-th
cell of π. If X is a graph we say a partition π of V (X) is equitable if there
are constants Bi,j such that any vertex in the i-th cell of π has exactly Bi,j
neighbours in the j-th cell. We view the matrix B as the adjacency matrix
of a directed graph with the cells of π as its vertices and with Bi,j directed
edges going from the i-th cell to the j-th cell. We denote this directed graph
by X/π.
Note that if π is equitable then the subgraph induced by each cell of π
is regular and the bipartite graph formed by the edges joining distinct cells
is semiregular. Thus a graph is regular if and only if the trivial partition is
equitable, while the discrete partition is always an equitable partition.
3.14.3 Theorem. Let X be a graph with adjacency matrix A and let π be
a partition of V (X) with characteristic matrix M . Then the following are
equivalent:
(a) π is equitable.
48
3.15. Some Algebra
(b) col(M ) is A-invariant.
(c) AM = M B, where B is the adjacency matrix of X/π.
If G is a group of automorphisms of X, then its orbits form an equitable
partition. If X is distance-regular then the distance partition δ with respect
to a given vertex is equitable, and the adjacency matrix of X/δ is the
intersection matrix of the graph. (This is more a matter of definition than
of argument.)
3.15
Some Algebra
Suppose X is a distance-regular graph with diameter d. The set of matrices
A = A0 , . . . , Ad span a vector space of dimension d over R, which we denote
by R[A]. It is not hard to show this vector space is closed under matrix
multiplication, and therefore it is a matrix algebra. (It is also closed under
Schur multiplication.)
Now multiplication by Ai is a linear map from R[A] to itself and, relative
to the basis A0 , . . . , Ad , we can represent this linear map by a (d+1)×(d+1)
matrix Bi . We say that Bi is the i-th intersection matrix. It is not hard to
show that the map
Ai 7→ Bi
extends to an isomorphism from R[A] to the algebra generated by the intersection matrices. Thus for example Ai and Bi have the same minimal
polynomial.
If we define the numbers ai , bi and ci as in Table ??, then we find that














bd−1 
0 k

1 a1 b1


c2 a2
B1 = 

...

b2
...
...
cd−1 ad−1
cd
ad
In particular B1 is tridiagonal. Since Ai = pi (A1 ) where p is a polynomial
of degree i in Ai , it follows that Bi = pi (B1 ). The matrices Ai and Bi have
the same set of eigenvalues, since they have the same minimal polynomial.
49
3. Real Lines
Since B1 is tridiagonal and bi ci+1 > 0 for i = 0, . . . , d − 1, its eigenvalues
are simple. (Proof?). Hence A1 has exactly d + 1 distinct eigenvalues, and
we can find them by computing the eigenvalues of B1 . (Although B is not
symmetric, it is diagonalisable and its eigenvalues are real.)
3.16
Inner Products
We assume familiarity with real and complex inner products. Note that
for us a complex inner product is linear in the second variable. Thus the
standard complex inner product on Cd is given by
hx, yi = x∗ y.
In addition to Rd and Cd , there are other inner product spaces which we
use.
Recall that M ◦ N denotes the Schur product of the two matrices M
and N (of the same order). Then
hA, Bi := tr(A∗ B) = sum A ◦ B
is an inner product of the space of complex m × n matrices. If we restrict
to real matrices, this inner product takes the simpler form
hA, Bi = tr(AT B) = sum(A ◦ B)
The Hermitian d × d matrices form a real vector space of dimension d2 ,
and this is again an inner product space,
contains the real symmetric
which
d+1
matrices as a subspace of dimension 2 .
If x1 , . . . , xn is a sequence of vectors in an inner product space, their
Gram matrix is the n × n matrix G such that
Gi,j = hxi , xj i.
According as our field is R or C, this is a symmetric or Hermitian matrix.
3.16.1 Lemma. If x1 , . . . , xn is a sequence of vectors in an inner product
space with Gram matrix G, then rk(G) is the dimension of the subspace
spanned by x1 , . . . , xn .
If Q is an orthogonal matrix, then the Gram matrix of x1 , . . . , xn and
of Qx1 , . . . , Qxn are equal. Conversely, if the Gram matrices of x1 , . . . , xn
and y1 , . . . , yn are equal, then there is an orthogonal matrix Q such that
Qxi = yi for all i. (Exercise.)
50
3.17. Positive Semidefinite Matrices
3.17
Positive Semidefinite Matrices
A matrix A is positive semidefinite if it is Hermitian and, for all x,
hx, Axi ≥ 0.
It is positive definite if it is positive semidefinite and
hx, Axi = 0
if and only if x = 0. A positive definite matrix must be invertible. The sum
of two positive semidefinite matrices is positive definite, and the sum of
of a positive definite and a positive semidefinite matrix is positive definite.
The identity matrix is a cheap example of a positive definite matrix. More
generally:
3.17.1 Lemma. A Gram matrix is positive semidefinite.
3.17.2 Lemma. A matrix A is positive definite if and only hx, Ayi is an
inner product.
3.17.3 Theorem. If A is Hermitian, the following statements are equivalent:
(a) A is positive semidefinite.
(b) The eigenvalues of A are nonnegative.
(c) A = U ∗ U for some matrix U .
(d) Each principal minor of A is nonnegative.
The most difficult step in proving the above theorem is in showing that
(d) implies (a). The matrix U in (b) can be chosen to be Hermitian, or
triangular.
One consequence of our theory is the Cauchy-Schwarz inequality. Suppose x and y are vectors in an inner product space with Gram matrix G.
Then det(G) ≥ 0. Since
!
hx, xi hx, yi
G=
hy, xi hy, yi
51
3. Real Lines
it follows that
hx, yihy, xi ≤ hx, xihy, yi.
Further, if equality holds then det(G) = 0 and therefore the space spanned
by x and y has dimension at most one, thus x and y are parallel.
52
Chapter 4
Complex Lines
We investigate the complex analogs of the results in the previous chapter.
4.1
The Absolute Bound
If x is a non-zero vector in Cd then the matrix
1
xx∗
x∗ x
represents orthogonal projection onto the line spanned by x. This is a
Hermitian matrix with rank one. If X and Y are projections onto complex
lines, we define the inner product
tr(X ∗ Y )
to be the squared cosine of the angle between the two lines. If x and y are
unit vectors and X = xx∗ and Y = yy ∗ , then
tr(X ∗ Y ) = hx, yihy, xi = |hx, yi|2 .
4.1.1 Theorem. A set of equiangular lines in Cd has size at most d2 .
Proof. Suppose X1 , . . . , Xn are the projections onto an equiangular set of
n lines in Cd with squared cosine γ. We show that these projections form
a linearly independent set in the vector space of Hermitian d × d matrices,
and deduce the bound from this.
53
4. Complex Lines
Assume that we have real scalars c1 , . . . , cn such that
0=
X
ci Xi .
i
If we take the inner product of both sides with Xr , on the left, we get
0 = (1 − γ)r +
X
ci γ,
i
from which we deduce that cr is independent of r and hence that
0=
X
Xi .
i
Since the trace of the right side is n, we have a contradiction and so we
conclude that X1 , . . . , Xn is linearly independent.
The set of d×d Hermitian matrices is a real vector space with dimension
2
d , and therefore n ≤ d2 as asserted.
4.2
The Relative Bound
Physicists are only interested in equiangular sets of size d2 ; we will consider
a broader class of problems.
Suppose X1 , . . . , Xn are the projections onto a set of n equiangular lines
with squared cosine γ, and that there are scalars c1 , . . . , cn such that
I=
X
ci X i .
i
Then taking the inner product with Xr as before, we deduce that cr is
independent of r, and hence that
X
n
Xi = I.
d
i
It follows that
1 − γ + nγ =
and so
γ=
n−d
d(n − 1)
When n = d2 , this yields that
γ=
54
n
d
1
.
d+1
4.3. Gram Matrices
4.2.1 Theorem. If there is an equiangular set of n lines in Cd with squared
cosine γ and dγ < 1, then
d − dγ
n≤
1 − dγ
and, if equality holds and X1 , . . . , Xn are the projections onto the lines,
then
X
n
Xi = I.
d
i
Proof. Exercise.
4.3
Gram Matrices
The Gram matrices of sets of equiangular lines in Cd do not lead to graphs
in general, but they still have some interesting properties.
Suppose x1 , . . . , xn are unit vectors spanning a set of equiangular lines
in Cd with squared cosine γ, let G be their Gram matrix and let S be the
matrix defined by
√
G = I + γS.
Thus S is a Hermitian matrix with zero diagonal and with all off-diagonal
entries having absolute value 1.
Assume now that the relative bound is tight, and let Z be the d × n
matrix with the vectors x1 , . . . , xn as its columns. Then
G = ZT Z
and
ZZ T =
n
I,
d
whence
n
G.
d
Thus the eigenvalues of G are 0 (with multiplicity n − d) and n/d (with
multiplicity d) and therefore the eigenvalues of S are
G2 =
1
−√ ,
γ
n−d
√
d γ
with respective multiplicities n − d and d.
In general the entries of S are not integers and thus we cannot argue,
as we did in the real case, that γ −1/2 must be an integer.
55
4. Complex Lines
4.4
Type-II Matrices
We use A ◦ B to denote the Schur product of two matrices with the same
order. If A ◦ B = J, we say that B is the Schur inverse of A, and write
B = A(−) .
A v × v complex matrix W is a type-II matrix if
W W (−)T = vI.
Note that if W is any Schur invertible v×v matrix, then the diagonal entries
of W W (−)T are all equal to v. Hadamard matrices are type-II matrices.
If W is type II, then so are W T and W (−) . If D is diagonal and invertible,
then DW and W D are both type II; if P is a permutation matrix then P W
and W P are type-II. If W1 and W2 are type-II matrices, so is their Kronecker
product W1 ⊗ W2 .
We say that a complex matrix is flat if all its entries have the same
absolute value. Hadamard matrices are flat.
4.4.1 Lemma. If W is a square complex matrix, then any two of the following imply the third:
(a) W is type II.
(b) A non-zero scalar multiple of W is unitary.
(c) W is flat.
Proof. Exercise.
Suppose x1 , . . . , xn is an equiangular set of n lines in Cd with squared
cosine γ and the relative bound is tight. Let G be the Gram matrix of a
set of unit vectors spanning these lines and set
1
S = √ (G − I).
γ
From the previous section, the eigenvalues of S are
1
τ := − √ ,
γ
56
θ :=
n−d
√
d γ
4.4. Type-II Matrices
and so (S − τ I)(S − θI) = 0 and hence
S 2 = (θ + τ )S − θτ I =
n − 2d
√ S + (n − 1)I.
d γ
4.4.2 Lemma. Suppose S is the Seidel matrix of an equiangular set of n
lines in Cd with squared cosine γ. If the relative bound is tight and
λ + λ−1 +
n − 2d
√ = 0,
d γ
then λI + S is a type-II matrix.
Proof. We note that
S = S ∗ = S (−)T
and therefore
(λI + S)(λI + S)(−)T = (λI + S)(λ−1 I + S (−)T )
= (λI + S)(λ−1 I + S)
= I + (λ + λ−1 )S + S 2
= I(1 − θτ ) + (λ + λ−1 + θ + τ )S.
The lemma follows immediately.
Let us call a matrix d-flat if its off-diagonal entries all have the same
absolute value and its diagonal entries all have the same absolute value.
We say that a d-flat matrix isnormalized if it off-diagonal entries all have
absolute value 1, and its diagonal entries are real.
4.4.3 Lemma. Suppose W is a normalized d-flat type-II matrix. If Wi,i =
δ 6= 1 and S := W − δI, then S is the Gram matrix of a set of equiangular
lines realizing the relative bound.
Proof. Assume W is v × v. We see that S (−)T = S ∗ and so
W (−)T = δ −1 I + S ∗ .
Therefore
vI = W W (−)T = I + δS ∗ + δ −1 S + SS ∗
(4.4.1)
57
4. Complex Lines
and on taking the conjugate-transpose of this, we get
vI = I + δS + δ −1 S ∗ + SS ∗ .
Comparing this with (4.4.1) yields that
(δ − δ −1 )S = (δ − δ −1 )S ∗ .
Therefore S is Hermitian and so (4.4.1) implies that
S 2 + (δ + δ −1 )S − (v − 1)I = 0.
If τ is the least eigenvalue of S, then
1
I+ S
τ
is positive semidefinite with one positive eigenvalue.
4.5
The Unitary Group
Let V be an inner product space. A linear operator M on V is orthogonal
if
hM u, M vi = hu, vi
for all u and v in V . If the inner product is complex we often call an
orthogonal operator unitary. The matrix that represents an orthogonal
operator relative to an orthogonal basis is also said to be orthogonal. The
adjoint M ∗ of M is the operator defined by the condition
hU, M vi = hM ∗ u, vi.
Thus M is unitary if and only if M ∗ = M −1 .
An orthogonal operator clearly preserves the length of a vector.
4.5.1 Lemma. A linear operator preserves length if and only if it is orthogonal.
Proof. We assume the inner product is complex, as this case is slightly
trickier and it implies the real case.
58
4.5. The Unitary Group
If M is a unitary linear operator on V and x, y ∈ V , then
hx + y, x + yi = hM x + M y, M x + M yi
= hM x, M xi + hM y, M yi + hM x, M yi + hM y, M xi
= hx, xi + hy, yi + hM x, M yi + hM y, M xi
Therefore
hx, yi + hy, xi = hM x, M yi + hM y, M xi
Setting ix in place of x in this identity yields
−ihx, yi + ihy, xi = −ihM x, M yi + ihM y, M xi
and therefore
hx, yi = hM x, M yi
for all x and y.
We construct a useful class of unitary mappings. Suppose ϕ ∈ V ∗ \ 0
and y ∈ V \ 0. If we define the mapping τ by
τ (x) := x + ϕ(x)y
then τ is linear and fixes each vector in ker ϕ. Then
hτ (x), τ (x)i = hx, xi + ϕ(x)hx, yi + ϕ(x)hy, xi + ϕ(x)ϕ(x)hy, yi
(4.5.1)
and if τ preserves length, then ϕ(x) = 0 whenever hy, xi = 0. Thus the
kernel of the linear map
x 7→ hy, xi
is contained in ker(ϕ). Since both kernels have codimension one in V , they
are equal and consequently there is a non-zero scalar λ such that
ϕ(x) = λhy, xi
for all x.
If we substitute φ(x) = −λhy, xi/hy, yi in (4.5.1) then
hτ (x), τ (x)i = hx, xi − (λ + λ − λλ)
hx, yihy, xi
hy, yi
and therefore τ preserves length if and only if
λ + λ − λλ = 0,
which happens if and only if k1 − λk = 1.
59
4. Complex Lines
4.5.2 Corollary. The map τ : V → V given by
τ (x) = x − 2
hy, xi
y
hy, yi
is unitary.
We note that
τ (y) = y − 2y = −y
and from this it follows that τ 2 = 1. The map τ is therefore called a complex
reflection.
4.6
An Extraspecial Group
A diagonal matrix is unitary if its diagonal entries have absolute value 1.
If D is a diagonal matrix and P a permutation matrix of the same order,
then P DP T = D0 is diagonal and so P D = D0 P . Define two d × d matrices
X and Y as follows. Let e0 , . . . , ed−1 be the standard basis for Cd , with the
understanding that the indices 0, . . . , d − 1 are integers modulo d, and set
θ = exp(2πi/d). Then
Xei = ei+1 ,
Y ei = θi ei .
Thus X is a permutation matrix and Y is diagonal. The Weyl-Heisenberg is
the group generated by X, Y and θI. (A physicist would call it a generalized
Pauli group.)
We investigate some of the properties of this group, which we denote by
G. We calculate
XY ei = θi ei+1 ,
Y Xei = θi+1 ei+1 ,
and thus
Y X = θXY,
in particular X and Y do not commute. It also follows from this relation
that each element of our group can be written in the form
θr X s Y t
60
4.6. An Extraspecial Group
where 0 ≤ r, s, t ≤ d − 1. This shows that |G| ≤ d3 . You should prove that
equality holds.
The subgroup if G consisting of the elements θr I is central with order
d. The quotient G/D is abelian and is isomorphic to Z2d .
4.6.1 Lemma. The group G acts irreducibly on Cd .
Proof. Suppose U is a non-zero subspace that is fixed by G, and let u be a
non-zero vector in it. Then U contains X r u for all r, and so we may assume
without loss that u1 6= 0. Then the vector
X
Y r u = ce1 ,
r
for some scalar c. Hence U contains 1 , and since er = X r e1 , we conclude
that U contains a basis for Cd . Therefore U = Cd , and so we have shown
that no proper subspace of Cd is G-invariant.
This lemma has important consequences. First, it implies that the only
matrices that commute with all elements of G are the scalar matrices. Second it implies that the subspace of Matd×d (C) spanned by the elements of
G has dimension d, and thus this subspace is Matd×d (C). (The second fact,
due to Burnside, implies the first, due to Schur.)
The Weyl-Heisenberg group can be used to construct sets of d2 equiangular lines in Cd . The idea is to choose a non-zero vector f in Cd , and
consider the lines spanned by the images of f under the action of the d3
elements of the group. This will produce at most d2 lines, and in certain
cases the result is a set of d2 equiangular lines. To be more specific, if d = 2
we make take f to be one of the two vectors
q

√ 
1  ± q
3± 3 
√
√
6 eiπ/4 3 ∓ 3
When d = 3 we make take f to be
 
1
1

√ 
1 .
2 0
There is no algorithm for finding f , but examples have been constructed in
dimensions 2 − 7 and 18. An example is also known when d = 8, although
it uses a different group. The physicists call f a fiducial vector.
61
Chapter 5
Mutually Unbiased Bases
Suppose x1 , . . . , xd and y1 , . . . , yd are two orthonormal bases of some inner
product space. We say these two bases are mutually unbiased if there is a
real scalar γ such that, for all i and j,
hxi , yj i|2 = γ.
We will call γ the squared cosine of the set of bases. A set of bases is
mutually unbiased if each pair from it is unbiased. The columns of the two
matrices
!
!
1 0
1 −1
,
0 1
−1 1
form a mutually unbiased pair of bases in R2 .
5.1
Basics
Suppose x1 , . . . , xd and y1 , . . . , yd are a pair of mutually unbiased bases. If
X
yj =
ci x i
i
then
1 = kyi k2 =
d
X
|ci |2 = dγ.
i=1
Therefore
1
γ= .
d
63
5. Mutually Unbiased Bases
The columns of the unitary matrices M and N form a mutually unbiased
pair of bases if and only if the columns of I and M −1 N do. Thus any set of
r mutually unbiased bases can be specified by a set of r unitary matrices,
one of which is the identity.
5.1.1 Lemma. If M is unitary and I and M are unbiased, then A is flat.
Recall that unitary matrix is a type-II matrix if and only if it is flat.
Thus each flat unitary matrix determines a mutually unbiased pair of bases.
5.2
Bounds
5.2.1 Theorem. A set of mutually unbiased bases in Cd contains at most
d + 1 bases; in Rd we have at most d2 + 1 bases.
Proof. Suppose we have vectors xi,j where 1 ≤ i ≤ m and for each i, the
vectors xi,1 , . . . , xi,d form an orthonormal basis. Assume further that these
bases are mutually unbiased. Let Xi,j denote the projection corresponding
to xi,j and let G be the Gram matrix of the projections. Then G has the
form
Imd + γ((Jm − Im ) ⊗ Jd ).
We determine the rank of G. Its eigenvalues are
γ(m − 1)d + 1, 1 − γd, 1
with respective multiplicities 1, m − 1 and md − m. As dγ = 1, we see
that rk(G) = md − d + 1. Hence the projections Xi,j span a subspace of
the space of Hermitian matrices with dimension md − m + 1 and so, in the
complex case,
md − m + 1 ≤ d2 ,
from which it follows that m ≤ d+1. In the real case we get m ≤ (d+2)/2.
5.3
MUB’s
If x1 , . . . , xd and y1 , . . . , yd are two orthonormal bases in Cd , we say that
they are unbiased if there is a constant γ such that for all i and j,
hxi , yj ihyj , xi i = γ.
64
5.3. MUB’s
In other words, the angle between any two lines spanned by vectors in
different bases is the same. A set of orthonormal bases is mutually unbiased
if each pair of bases in it is unbiased. If U and V are d × d unitary matrices
then their columns provide a pair of orthonormal bases, and these bases
are unbiased if and only if the matrix U ∗ V is flat. Note that U ∗ V is itself
unitary, and that its columns and the standard basis of Cd are unbiased.
The two bases
!
!
0
1
,
;
1
0
!
!
1
1 1
1
√
,√
2 1
2 −1
are mutually unbiased.
The angle between lines corresponding to vectors from distinct orthogonal bases is determined by d. To see this, suppose x1 , . . . , xd and y1 , . . . , yd
are orthogonal and unbiased with |hxi , yj i|2 = γ. Then since x1 , . . . , xd is
an orthonormal basis
X
y1 = hxi , y1 ixi
and
hy1 , y1 i =
X
|hxi , y1 i|2 = dγ.
i
Hence γ = d−1 (and |hxi , yj i| = d−1/2 ).
Our goal is to find mutually unbiased sets of bases with maximal size.
How large can a mutually unbiased set of bases be? If P and Q are projections onto lines spanned by two vectors from a set of mutually unbiased
bases, then hP, Qi is 0, 1 or d−1 . The Gram matrix of the projections onto
lines spanned by vectors from a set of mutually unbiased bases is
G = Im ⊗ Id +
1
(Jm − I) ⊗ Jd .
d
We determine the rank of G by counting its nonzero eigenvalues. The
eigenvalues of (Jm − I) ⊗ Jd are
eigenvalue
(m − 1)d
0
−d
multiplicity
1
m(d − 1)
m−1
65
5. Mutually Unbiased Bases
Thus the eigenvalues of I + d1 (Jm − I) ⊗ Jd are
eigenvalue
m
1
0
multiplicity
1
m(d − 1)
m−1
Thus rk(G) = 1 + md − m and therefore
1 + md − m ≤ d2 ,
from which it follows that m ≤ d + 1.
Note. If we work in Rd we get
1 + md − m ≤
d2 + d
2
and then we find that m ≤ 1 + d2 .
The columns of a unitary matrix form an orthonormal basis. In fact a
matrix H is unitary if and only if its columns form an orthonormal basis.
Suppose H and K are unitary then the columns of H and K are unbiased
if and only if all entries of H ∗ K have absolute value √1d . So H ∗ K is flat and
since it is a product of unitary matrices it is unitary. Note that H and K
are unbiased if and only if I and H ∗ K are. Thus each flat unitary matrix
gives a pair of unbiased bases in Cd (matrix, identity).
Suppose the columns of matrices H1 , . . . , Hm and K1 , . . . , Km form mutually unbiased bases in Cd and Ce respectively. Then the Kronecker products
Hi ⊗ Ki
give a set of m mutually unbiased bases in Cde . (This is very easily verified.)
It follows that in any dimension there is a set of at least three mutually
unbiased bases.
5.4
Real MUB’s
We briefly consider the real case. This received almost no attention prior
to the physicists’ work on the complex case.
We note first that a flat orthogonal matrix is a scalar multiple of a
Hadamard matrix. It follows that if we have a real pair of mutually unbiased
matrices in Rd then either d = 2 or 4 | d.
66
5.4. Real MUB’s
5.4.1 Lemma. If there is a set of three mutually unbiased bases in Rd , then
d is an even square.
Proof. Suppose H and K are d × d Hadamard matrices such that the
columns of
1
1
I, √ H, √ K
d
d
are mutually unbiased. Then
1 T
H K
d
must be a flat real orthogonal matrix and therefore
1
√ HT K
d
is a Hadamard matrix. This implies that
√
d must be rational.
5.4.2 Lemma. If there is a set of four mutually unbiased bases in Rd , then
16 | d.
Proof. Suppose we have four mutually unbiased bases in Rd , the first of
which is the standard basis, and assume that d = 4s2 . Then the last three
bases come from three Hadamard matrices H, K and L such that if x, y
and z respectively are columns from these three matrices, then
hx, yi = hx, zi = hy, zi = 2s.
We consider the equation
h1, (x + y) ◦ (x + z)i = hx + y, x + zi.
Since x, y and z are ±1 vectors, the entries of x + y and x + z are 0 and
±2 and therefore the left side above is divisible by 4. On the other hand
hx + y, x + zi = hx, xi + hx, yi + hx, zi + hy, zi = 4s2 ± 2s ± 2s ± 2s
and therefore s must be even.
67
5. Mutually Unbiased Bases
5.5
Cayley Graphs and Incidence
Structures
Let G be an abelian group and suppose D ⊆ G. The Cayley graph X(G, D)
has the elements of G as its vertices, and (g, h) is an arc if hg −1 ∈ D. Any
character ψ of G is an eigenvector for A(X), with eigenvalue ψ(D).
The restriction ψD lies in Cd , and if ψ and ϕ are characters of G, then
hψ D, ϕ Di = (ψϕ−1 )D.
Since the product ψϕ−1 is a character of G, we see that the above inner
product is an eigenvalue of X(G, D). In particular the squared cosine of the
angle between the complex lines spanned by ψ D and ϕD is the absolute
value of an eigenvalue of X(G, D).
If D ⊆ G then we can view the translates Dg, where g ∈ G, as the
blocks of an incidence structure. If N is the adjacency matrix of X(G, D),
then the adjacency matrix of the incidence graph of this incidence structure
can be taken to be
!
0 N
A=
.
NT 0
Then
!
NNT
0
2
A =
.
0
NT N
Now since G is abelian, N N T = N T N and hence if θ is an eigenvalue of A,
then θ2 is an eigenvalue of N N T .
On the other hand, since N is normal, there is an orthogonal basis of
v
C that consists of common eigenvectors of N and N T . If
N z = λz
then
N T z = λz
and z is an eigenvector for N N T with eigenvalue λλ. It follows that
θ = ±|λ|.
Since an incidence graph is bipartite, its spectrum is symmetric about zero,
and we conclude that the number of negative eigenvalues of the incidence
graph is equal to the number of different values taken by the squared cosine
of the angles between the v lines in Cd corresponding to the characters of
G.
68
5.6. Difference Sets
5.6
Difference Sets
We apply the machinery developed in the previous section to construct sets
of d2 − d + 1 equiangular lines in Cd . While this is interesting in its own
right, it also serves as a warm up for the more difficult task of constructing
mub’s.
The group algebra of the group G over C consists of all sums
X
cg g
g
where only finitely many of the coefficients cg are not zero. (As our groups
will be finite, this restriction will not be an issue.) We add and multiply
these sums in the obvious fashion. If D ⊆ G, we identify D with the element
X
d
d∈D
of the group algebra. We also use D−1 to denote
X
d−1 .
d∈D
Then DD−1 can be viewed as the multiset of differences gh−1 , where g and
h run over the elements of D. (One advantage of the group algebra setup
is that we can avoid reference to multisets.)
A subset D of G is a difference set if there is an integer λ such that
DD−1 = |D|1G + λ(G − 1G ).
The difference set is parameterized by the triple
(|G|, |D|, λ);
we call λ the index of the difference set. This definition of difference set is
consistent with the one we used in ??, except that there we used abelian
groups with addition (rather than multiplication) as the group operation.
If D ⊆ G, then we can form an incidence structure with the elements
of G as its points and the translates Dg (for g in G) as its blocks. In the
cases of interest to us there will be |G| distinct translates of D.
69
5. Mutually Unbiased Bases
5.6.1 Lemma. If D is a difference set in the group G with index λ, then
the associated incidence structure is a symmetric design with parameter set
(|G|, |D|, λ).
Proof. Exercise.
The incidence matrix of the incidence structure is the adjacency matrix
of the Cayley graph X(G, D).
5.7
Difference Sets and Equiangular Lines
Suppose D is a difference set with index λ in the abelian group G and
assume d = |D| and v = |G|. Then the vectors
ψ D,
where ψ runs over the characters of G, span a set of v lines in Cd . To
determine the angles, we need the values |ψ(D)|. We have
|ψ(D)|2 = ψ(D)ψ(D) = ψ(DD−1 )
and since
DD−1 = d1G + λ(G − 1G )
it follows that
|ψ(D)|2 = d − λ + λψ(G).
If ψ is the trivial character, then ψ(D) = d and ψ(G) = v and consequently
v =1+
d2 − d
.
λ
If ψ is not trivial, then ψ(G) = 0 and
|ψ(D)|2 = d − λ.
This implies that the restrictions ψD form a set of equiangular lines in Cd .
In particular, when λ = 1 we obtain a set of d2 − d + 1 equiangular lines in
Cd . It can be shown that this set of lines meets the relative bound.
Since the values of a character of an abelian group are roots of unity, the
set of lines we obtain from a difference set are spanned by flat vectors. And
since he characters form a group, these vectors form a group under Schur
multiplication. (This group is a homomorphic image of G. Is it isomorphic
to G?)
70
5.8. Relative Difference Sets and MUB’s
5.8
Relative Difference Sets and MUB’s
Let G be a group with a normal subgroup N . A subset D of G is a relative
difference set if
DD−1 = |D|1G + λ(G − N ).
(5.8.1)
Thus a difference set relative to the identity subgroup is a difference set as
before.
We offer a relevant example. Let F be a field of odd order q, let G be
the vector space of dimension two over F and let N be the subgroup (0, F)
of G. Then
D = {(x, x2 ) : x ∈ F}
is a difference set relative to N with index 1.
The defining equation for a relative difference set implies that no two
elements of D lie in the same coset of N , and thus wwe have the bound
|D| ≤ |G : N |.
A relative difference set is semiregular if equality holds in this bound. Only
semiregular relative difference sets will be of interest to us.
If we set d equal to |D| and apply the trivial character to each side of
(5.8.1), we find that
d2 = d + λ|N |(|G : N | − 1)
and consequently if D is semiregular, then
d = λ|N |,
|G| = λ|N |2 .
We can divide the characters of G into three classes:
(a) The trivial character ψ, for which ψ(G) = |G|.
(b) Non-trivial characters ψ such that ψ N is trivial, where ψ(G) = 0 and
ψ(N ) = 0.
(c) Non-trivial characters ψ such that ψ N is not trivial, where ψ(G) and
ψ(N ) are both zero.
We note that the characters whose restriction to N is trivial form a
subgroup N∗ of the character group G∗ , isomorphic to (G/N )∗ . The corresponding values of |ψ(D)|2 are
71
5. Mutually Unbiased Bases
(a) |D|2 ,
(b) |D| − λ|N | = 0,
(c) |D|.
If ϕ and ψ are characters of G, then ϕ D and ψ D are orthogonal if and
only if ϕψ −1 N is trivial. Hence the characters in a given coset of N∗ form
an orthogonal basis of Cd , and so we obtain as set of d/λ mutually unbiased
bases. We may adjoin the standard basis to this set, thus arriving at a set
of 1 + λ−1 d mub’s in Cd .
5.9
Type-II Matrices over Abelian Groups
Let N be a group. If W is a matrix with entries from N , we define W (−)
to be the matrix with the same order such that
−1
(W (−) )i,j = Wi,j
.
A type-II matrix over N is a v × v matrix W with entries from N such that
W W (−)T = vI + λN (J − I).
If we apply the trivial character of N to both sides, we obtain the equation
J 2 = vI + λ|N |(J − I),
whence we have
v = λN.
Suppose N is abelian. If ψ is a character of N , let Wψ denote the matrix
we get by applying ψ to the entries of W . Then ψ(N ) = 0 if ψ is not trivial
and
Wψ Wψ∗ = vI.
Thus Wψ is a flat type-II matrix. If ϕ is also a character of N , then
Wψ ◦ Wϕ = Wψϕ
and so the matrices Wψ form a group of order |N | under Schur multiplication.
72
5.10. Difference Sets and Equiangular Lines
5.10
Difference Sets and Equiangular Lines
5.10.1 Lemma. Let G be an abelian group of order n, and let ψ1 , . . . , ψn
be the characters of G. Suppose N is a 01-group matrix over G. If hT is
a row of N , then the number of angles between the lines spanned by the
vectors h ◦ ψr is one less than the number of eigenvalues of N N T .
Proof. If ψ and ϕ are characters of G, then
hh ◦ ψ, h ◦ ϕi = hh, ψ ◦ ϕi
(5.10.1)
If χ is a character for G, then N χ = λχ for some λ and therefore, since the
entries of χ are complex numbers of norm 1,
|hh, χi| = |λ|.
So
|hh ◦ ψ, h ◦ ϕi|2
is equal to the eigenvalue of N N T on ψϕ.
Note that if the weight of the vector h above is d, then the vectors h ◦ ψ
lie a d-dimensional subspace of Cn .
If X is the incidence graph the design, then
0 N
A(X) =
NT 0
and
!
!
NNT
0
A(X) =
.
T
0
N N
It follows that number of angles is equal to the number of non-negative
eigenvalues of X.
If N is a group matrix over Zn2 +n+1 and an incidence matrix for a
projective plane of order n, then
2
N N T = nI + J
which has eigenvalues (n + 1)2 and n (with multiplicity 1 and n2 + n respectively). Hence we obtain a set of n2 + n + 1 equiangular lines in Cn+1
whenever n is a prime power. The size of this set of lines meets the relative
bound, as you are invited to prove.
A complex matrix is flat if all its entries have the same absolute value.
The vectors spanning the d2 − d + 1 lines are flat; it can be shown that a
set of flat equiangular lines in Cd has size at most d2 − d + 1.
73
5. Mutually Unbiased Bases
5.11
Affine Planes
Let V be a vector space of dimension two over GF (q); we write it elements
as pairs (x, y). Let [a, b] denote the set of points
{(x, y) : y = ax + b}.
This a line in the affine plane over GF (q), and as we vary a and b we get all
lines except those parallel to the y-axis—the lines with infinite slope. It is
easy to verify that this structure is a divisible semisymmetric design. Our
problem is to show that there is an abelian group of automorphisms acting
regularly on points and lines.
There are two obvious sets of automorphisms. Let Tu,v : V → V be
given by
Tu,v (x, y) = (x + u, y + v).
We call the maps Tu,v translations, they form an abelian group of order q 2 .
If (x, y) is on the line [a.b], then
y + v − (a(x + u) + b) = (y − ax − b) − (au − v)
and therefore Tu,v (x, y) is on [a, b − au + v]. Thus we can define the image
of [a, b] under Tu,v to be [a, b − au + v], and with this definition Tu,v is
an automorphism of our incidence structure. We see that translations are
automorphisms that each line to a parallel line. In particular you may show
that the group of translations has q orbits on lines.
We can also define dual translations Su,v by
Su,v [a, b] = [a + u, b + v].
Then
y − (a + u)x − (b + v) = y + ux + v − ax − b
and so Su,v maps lines on (x, y) to the lines on (x, y + ux + v). Again we
get a group of automorphisms, with q orbits on points.
What we need though is an abelian group with one orbit on points and
one orbit on lines. Define
Ru,v = Tu,v Su,0 .
Then these q 2 automorphisms from an abelian group of order q 2 that acts
transitively on point and on lines. Consequently we get a set of q mutually
unbiased bases in Cq , that are all unbiased relative to the standard basis.
74
5.12. Products
This construction does not make use of the fact that finite fields are
associative, and we may use a commutative semifield in place of a field. All
known examples of mub’s can be constructed in this way.
5.12
Products
If H1 , . . . , Hm is a set of unitary matrices describing a set of mub’s in Cd
and K1 , . . . , Km is a second set giving mub’s in Ce , then the products
Hi ⊗ Ki ,
(i = 1, . . . , m)
give us a set of mub’s in Cde . This may not seem to be a very efficient
construction, but in many cases it is the best we can do. If d is a prime
power then there is a mutually unbiased set of bases of size d+1 in Cd ; hence
this product construction guarantees the existence of a mutually unbiased
set of three bases in any dimension. When d ∼
= 2 modulo four, there is no
better bound known in general.
There is one construction, due to Beth and Wocjan, which is better than
the product construction in some cases. Suppose we have an OA(k, q) and
a flat unitary matrix H of order q × q. Our array can be viewed as an
incidence structure with q 2 lines and kq points. Let M be the incidence
matrix of the dual; this has order q 2 × kq. Then the kd2 vectors
(1 ⊗ Hei ) ◦ M ej
2
form k mutually unbiased bases in Cq . If q = 26, then the product construction provides five mub’s in C576 . There is an OA(26, 6), and so we
obtain six mub’s in C676 .
75
Bibliography
[1] Sougato Bose. Quantum communication through an unmodulated spin
chain. Physical Review Letters, 91(20), November 2003.
[2] Matthias Christandl, Nilanjana Datta, Tony Dorlas, Artur Ekert, Alastair Kay, and Andrew Landahl. Perfect transfer of arbitrary states in
quantum spin networks. Physical Review A, 71(3):12, March 2005.
[3] David A. Cox. Galois Theory. Wiley-Interscience, Hoboken, NJ, 2004.
[4] Douglas R. Farenick. Algebras of Linear Transformations. Springer,
New York, 2001.
[5] Edward Farhi and Sam Gutmann. Quantum computation and decision
trees. Physical Review A, 58(2):915–928, August 1998.
[6] Albrecht Fröhlich and Martin J. Taylor. Algebraic Number Theory. Cambridge University Press, 1993.
[7] Alastair Kay. The Basics of Perfect Communication through Quantum
Networks. arxiv, page 9, February 2011.
[8] Cristopher Moore and Alexander Russell. Quantum Walks on the Hypercube. April 2001.
[9] W. A. Stein, , et al. Sage Mathematics Software.
77
Index
commutative semifield, 75
dual translations, 74
fidelity, 16
flat, 73
flat matrix, 6
instantaneous uniform mixing, 6
mutually unbiased, 65
perfect state transfer, 6
periodic at u, 17
periodic graph, 17
phase factor, 3
phase factors, 29
quantum state, 3
spectral decomposition, 7
spectral density, 16
squared cosine, 30
translations, 74
unbiased, 64
uniform mixing, 6
79